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CORNELL;. 
UNIVERSITY 
LIBRARY  V 


BOUGHT  WITH  THE  INCOME 
OF  THE  SAGE  ENDOWMENT 
FUND     GIVEN     IN    189I     BY 

HENRY  WILLIAMS  SAGE 


Cornell  University  Library 

QB  43.M927I6  1916 


An  introduction  to  astronomy, 


3  1924  012  499  756 


AN  INTRODUCTION  TO  ASTEONOMY 


THE  MACMILLAN  COMPANY 

NEW  YORK    ■    BOSTON  •    CHICAGO  -    DALLAS 

ATLANTA   •    SAN    FRANCISCO 

MACMILLAN  &   CO.,  Limited 

LONDON  •  BOMBAY  •  CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  Ltd. 

TORONTO 


Cornell  University 
Library 


The  original  of  tliis  book  is  in 
tine  Cornell  University  Library. 

There  are  no  known  copyright  restrictions  in 
the  United  States  on  the  use  of  the  text. 


http://www.archive.org/details/cu31924012499756 


AN  INTRODUCTION 


TO 


ASTRONOMY 


BY 


FOREST   RAY   MOULTON,   Ph.D. 

PKOFESSOR    OP    ASTRONOMY    IN    THE    UNIVERSITY    OF    CHICAGO 

RESEARCH    ASSOCIATE    OF    THE    CAKNBGIE    INSTITUTION 

OF  WASHINGTON 


NEW  AND  REVISED  EDITION 


Weto  goris 

THE   MACMILLAN   COMPANY 

1916 

All  rights  reserved 


OOPYEIGHT,    1906  AND   1916, 

By  TBE  MACMILLAN  COMPANY. 


Set  up  and  electrotyped.  Published  April,  1906.  Reprinted 
November,  1907;  July,  igo8;  April,  1910;  April,  1911;  September, 
1912;  September,  1913;  Optober,  1914. 

New  and  revised  edition  November,  igi6. 


J.  S.  Gushing  Co.  — Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

The  necessity  for  a  new  edition  of  "  An  Introduction  to 
Astronomy"  has  furnished  an  opportunity  for  entirely  re- 
writing it.  As  in  the  first  edition,  the  aim  has  been  to  pre- 
sent the  great  subject  of  astronomy  so  that  it  can  be  easily 
comprehended  even  by  a  person  who  has  not  had  extensive 
scientific  trainiag.  It  has  been  assumed  that  the  reader  has 
no  intention  of  becoming  an  astronomer,  but  that  he  has  an 
interest  in  the  wonderful  universe  which  surrounds  him,  and 
that  he  has  arrived  at  such  a  stage  of  iatellectual  development 
that  he  demands  the  reasons  for  whatever  conclusions  he  is 
asked  to  accept.  The  first  two  of  these ,  assumptions  have 
largely  determined  the  subject  matter  which  is  presented  ; 
the  third  has  strongly  influenced  the  method  of  presenting  it. 

While  the  aims  have  not  changed  materially  since  the  first 
edition  was  written,  the  details  of  the  attempt  to  accomplish 
them  have  undergone  many,  and  in  some  cases  important, 
modifications.  For  example,  the  work  on  reference  points  and 
liues  has  been  deferred  to  Chapter  IV.  If  one  is  to  know  the 
sky,  and  not  simply  know  about  it,  a  knowledge  of  the  coordi- 
nate systems  is  indispensable,  but  they  always  present  some 
difficulties  when  they  are  encountered  at  the  beginning  of  the 
subject.  It  is  believed  that  the  present  treatment  prepares 
so  thoroughly  for  their  study  and  leads  so  naturally  to  them 
that  their  mastery  will  not  be  found  dif&cult.  The  chapter  on 
telescopes  has  been  regretfully  omitted  because  it  was  not 
necessary  for  understanding  the  remainder  of  the  work,  and 
because  the  space  it  occupied  was  needed  for  treatrag  more 
vital  parts  of  the  subject.  The  numerous  discoveries  in  the 
sidereal  universe  during  the  last  ten  years  have  made  it  neces- 
sary greatly  to  enlarge  the  last  chapter. 


VI  PREFACE 

As  now  arranged,  the  first ,  chapters  are  devoted  to  a  discus- 
sion of  the  earth  and  its  motions.  They  present  splendid 
examples  of  the  characteristics  and  methods  of  science,  and 
amply  illustrate  the  care  with  which  scientific  theories  are 
established.  The  conclusions  which  are  set  forth  are  bound  up 
with  the  development  of  science  from  the  dawn  of  recorded 
history  to  the  recent  experiments  on  the  rigidity  and  the  elas- 
ticity, of  the  earth.  They  show  how  closely  various  sciences 
are  interlocked,  and  how  much  an  understanding  of  the  earth 
depends  upon  its  relations  to  the  sky.  They  lead  naturally  to 
a  more  formal  treatment  of  the  celestial  sphere  and  a  study  of 
the  constellations.  A  familiarity  with  the  brighter  stars  and 
the  more  conspicuous  constellations  is  regarded  as  important. 
One  who  has  become  thoroughly  acquainted  with  them  will 
always  experience  a  thrill  when  he  looks  up  at  night  into  a 
cloudless  sky. 

The  chapter  on  the  sun  has  been  postponed  until  after  the 
treatment  of  the  moon,  planets,  and  comets.  The  reason  is 
that  the  discussion  of  the  sun  necessitates  the  introduction  of 
many  new  and  difficult  topics,  such  as  the  conservation  of  en- 
ergy, the  disintegration  of  radioactive  elements,  and  the  prin- 
ciples of  spectrum  analysis.  Then  follows  the  evolution  of 
the  solar  system.  In  this  chapter  new  and  more  serious  de- 
mands are  made  on  the  reasoning  powers  and  the  imagination. 
Its  study  in  a  measure  develops  a  point  of  view  and  prepares 
the  way  for  the  consideration,  in  the  last  chapter,  of  the  tran- 
scendental and  absorbingly  interesting  problems  respecting 
the  organization  and  evolution  of  the  sidereal  universe. 

Lists  of  problems  have  been  given  at  the  ends  of  the  prin- 
cipal divisions  of  the  chapters.  They  cannot  be  correctly 
answered  without  a  real  comprehension  of  the  principles  which 
they  involve,  and  in  very  many  cases,  especially  in  the  later 
chapters,  they  lead  to  important  supplementary  results.  It  is 
strongly  recommended  that  they  be  given  careful  consideration. 

The  author  is  indebted  to  Mr.  Albert  Barnett  for  the  new 
star  maps  and  the  many  drawings  with  which  the  book  is  illus- 
trated, with  the  exception  of  Tigs.  23  and  30,  which  were 


PREFACE  VU 

kindly  furnished  by  Mr.  George  Otis.  He  is  indebted  to 
Professor  David  Eugene  Smith  for  photographs  of  Newtcta, 
Kepler,  Herschel,  Adams,  and  Leverrier.  He  is  indebted  to 
the  Lick,  Lowell,  Solar,  and  Yerkes  observatories  for  a  large 
amount  of  illustrative  material  which  was  very  generously 
furnished.  He  is  under  deeper  obligations  to  his  colleague, 
Professor  W.  D.  MacMillan,  than  this  brief  acknowledgment 
can  express  for  assistance  on  the  manuscript,  on  the  proofs, 
and  in  preparing  the  many  problems  which  appear  in  the  book. 

F.   E.    MOULTON. 
The  Uniteksitt  of  Chicago, 
September  25,  1916. 


CONTENTS 


CHAPTER  I 
Preliminary  Considerations 


1.  Science 

2.  The  value  of  science 

3.  The  origin  of  science  ... 

4.  The  methods  of  science 

5.  The  imperfections  of  science 

6.  Great  contributions  of  astronomy  to  science 

7.  The  present  value  of  astronomy  . 

8.  The  scope  of  astronomy 


PAGE 

1 

2 

4 

6 

10 

14 

16 

19 


CHAPTER   U 

THE   EARTH 

I.     The  Shape  op  the  Earth 

9.   Astronomical  problems  respecting  the  earth 
10,  11.   Proofs  of  the  earth's  sphericity 
12,  14,  15.  Proofs  of  the  earth's  oblateness 
1-3.    Size  and  shape  of  the  earth 

16.  The  theoretical  shape  of  the  earth 

17.  Different  kinds  of  latitude  . 

18.  Historical  sketch  on  the  shape  of  the  earth 


26 
27 
31 
33 
38 
39 
40 


II.     The  Mass  of  the  Earth  and  the  Conditioi«  of  its 
Interior 

19.  The  principle  by  which  mass  is  determined         ....      43 

20.  The  mass  and  density  of  the  earth      .        .        .        .        .        .45 

21-23.   Method?  pf  determining  the  density  of  the  earth  ...      46 


CONTENTS 


24.  Temperature  and  pressure  in  the  earth's  interior 

25,  26.   Proofs  of  the  earth's  rigidity  and  elasticity  . 

27.   Historical  sketch  on  the  mass  and  rigidity  of  the  earth 


PAGE 

51 
52 
62 


III.     The  Earth's  Atmosphere 

28.  Composition  and  mass  of  the  earth's  atmosphere 
29-31.   Methods  of  determining  height  of  the  atmosphere 

32.  The  kinetic  theory,  of  gases  .... 

33.  The  escape  of  atmospheres 

34.  Effects  of  the  atmosphere  on  climate  . 

35.  Importance  of  the  constitution  of  the  atmosphere 

36.  R61e  of  the  atmosphere  in  life  processes 

37.  Kefraction  of  light  by  the  atmosphere 

38.  The  twinkling  of  the  stars  .        .  .        . 


64 
65 


71 
72 
74 
74 
76 


CHAPTER  III 

THE   MOTIONS   OF  THE   EARTH 
I.     The  Rotation  of  the  Earth 

39.  The  relative  rotation  of  the  earth 

40.  The  laws  of  motion 

41-43.   Proofs  of  the  earth's  rotation 

44.  Consequences  of  the  earth's  rotation  . 

45.  Uniformity  of  the  earth's  rotation 

46.  The  variation  of  latitude 

47.  The  precession  of  the  equinoxes  and  nutation 


77 
79 
82 
85 
87 
89 
92 


II.     The  Revolution  of  the  Earth 

48.  Relative  motion  of  the  earth  with  respect  to  the  sun 
49-52.   Proofs  of  the  revolution  of  the  earth 

53.  Shape  of  the  earth's  orbit 

54.  Motion  of  the  earth  in  its  orbit    . 

55.  Inclination  of  the  earth's  orbit    . 

56.  The  cause  of  the  seasons     .... 
.57.  Relation  of  altitude  of  pole  to  latitude  of  observer 

58.  The  sun's  diurnal  circles 

59.  Hours  of  sunlight  in  different  latitudes 

60.  The  lag  of  the  seasons 

61.  Effect  of  eccentricity  of  earth's  orbit  on  seasons 

62.  Historical  sketch  of  the  motions  of  the  earth 


96 
98 
102 
103 
105 
107 
108 
109 
HI 
112 

u;j 

115 


CONTENTS 


XI 


ARTS. 

63. 
64. 
65. 
66. 
67. 
68. 


CHAPTER   IV 


EErBRENCE  Points  and  Lines 


Object  and  character  of  reference  points  and  lines 

The  geographical  system   . 

The  horizon  system  . 

The  equator  system  . 

The  ecliptic  system    . 

Comparison  of  systems  of  coSrdinates 
69,  70.   Finding  the  altitude  and  azimuth 
71,  72.   Finding  the  right  ascension  and  declination 
73.   Other  problems  of  position         .... 


PAGE 

121 
122 
123 
125 
127 
127 
130 
133 
135 


CHAPTER  V 

The  Constellations 

74.  Origin  of  the  constellations 138 

75.  Naming  the  stars 138 

76.  Star  catalogues 141 

77.  The  magnitudes  of  the  stars 142 

78.  The  first-magnitude  stars 143 

79.  Number  of  stars  in  first  six  magnitudes 145 

80.  Motions  of  the  stars 145 

81.  The  Milky  Way,  or  Galaxy 146 

82.  The  constellations  and  their  positions  (Maps)  ....  148 

83.  Finding  the  pole  star 149 

84.  Units  for  estimating  angular  distances 150 

85-101.   Ursa  Major,  Cassiopeia,  Locating  the  equinoxes,  Lyra, 

Hercules,  Scorpius,  Corona  Borealis,  Bootes,   I^eo,  An- 
dromeda, Perseus,  Auriga,  Taurus,  Orion,  Canis  Major, 

Canis  Minor,  Gemini 150 

102.   On  becoming  familiar  with  the  stars 167 


CHAPTER  VI 

Time 

103.  Definitions  of  equal  intervals  of  time 169 

104.  The  practical  measure  of  time   . 170 

105.  Sidereal  time 171 


Xll 


CONTENTS 


AET8.  ■"*«« 

106.  Solar  time l'i'2 

107.  Variations  in  length  of  solar  days     .        .                ...  172 

108.  Mean  solar  time 175 

109.  The  equation  of  time l'?6 

110.  Standard  time    ...                177 

111.  Distribution  of  time 179 

112.  Civil  and  astronomical  days 181 

113.  Place  of  change  of  date 181 

114-116.   Sidereal,  anomalistic,  and  tropical  years    ....  183 

117.  The  calendar 184 

118.  Finding  the  day  of  week  on  any  date 185 


CHAPTER   VII 


The   Moon 

119.  The  moon's  apparent  motion  among  the  stars 

120.  The  moon's  synodical  and  sidereal  periods 

121.  The  phases  of  the  moon     . 

122.  The  diurnal  circles  of  the  moon 
123  The  distance  of  the  moon  .... 

124.  The  dimensions  of  the  moon 

125,  126.   The  moon's  orbit  with  respect  to  earth  and  sun 

127.  The  mass  of  the  moon        .... 

128.  The  rotation  of  the  moon  .... 

129.  The  librations  of  the  moon 

130.  The  density  and  surface  gi-avity  of  the  moon 

131.  The  question  of  tl\e  moon's  atmosphere     . 

132.  Light  and  heat  received  from  the  moon     . 

133.  The  temperature  of  the  moon  . 
134-138.  The  surface  of  the  moon  . 
139.  Effects  of  the  moon  on  the  earth 
140-142.  Eclipses  of  the  moon  and  sun     . 


188 
189 
190 
192 
194 
196 
197 
198 
200 
201 
202 
203 
204 
205 
207 
217 
218 


CHAPTER  VIII 

THE   SOLAR   SYSTEM 

I.     The  Law  of  Gravitation 

143.  The  members  of  the  solar  system 

144.  Relative  dimensions  of  the  planetary  orbits 


226 

227 


CONTENTS 


xui 


AETS.  Page 

145.  Kepler's  laws  of  motion 229 

146,  147.   The  law  of  gravitation                230 

148.  The  conic  sections 234 

149.  The  question  of  other  laws  of  force  ....;.  236 

150.  Perturbations 237 

151.  The  discovery  of  Neptune 238 

152.  The  problem  of  three  bodies 241 

158.   Cause  of  the  tides 242 

154.  Masses  of  celestial  bodies 244 

155.  surface  gravity  of  celestial  bodies 245 


11.     Orbits,  Dimensions,  and  Masses  of  the  Planets 

156.  Finding  the  dimensions  of  the  solar  system 

157.  Elements  of  the  orbits  of  the  planets  (Table)    . 

158.  Dimensions  and  masses  of  the  planets  (Table) 

159.  Times  for  observing  the  planets 

160.  The  planetoids 

161.  The  question  of  undiscovered  planets 

162.  The  zodiacal  light  and  the  gegenscheiu 


246 
248 
252 
255 
257 
261 
262 


CHAPTER  IX 

THE  PLANETS 

I.     Mercurt  and  Venus 

163.  Phases  of  Mercury  and  Venus 266 

164.  Albedoes  and  atmospheres  of  Mercury  and  Venus    .        .        .  268 

165.  Surface  markings  and  rotation  of  Mercury        ....  269 

166.  The  seasons  of  Mercury 270 

167.  Surface  markings  and  rotation  of  Venus 271 

168.  The  seasons  of  Venus 272 


II.     Mars 


169.  The  satellites  of  Mars 

170.  The  rotation  of  Mars 

171.  The  albedo  and  atmosphere  of  Mars 

172.  The  polar  caps  and  temperature  of  Mars 

173.  The  canals  of  Mars    .... 

174.  Explanations  of  the  canals  of  Mars  . 


273 
274 
276 

277 
283 
285 


XIV 


CONTENTS 


III.     Jupiter 


176,  176.   Jupiter's  satellite  system    .        .        .        . 

177,  Discovery  of  the  velocity  of  light 

178,  179.   Surface 'markings  and  rotation  of  Jupiter  . 
180.   Physical  condition  and  seasons  of  Jupiter 


PAGE 

289 
291 
292 
296 


IV.     Saturn 

181.   Saturn's  satellite  system    .... 
182-184.    Saturn's  ring  system  . 
186.    Surface  markings  and  rotation  of  Saturn  . 
186.   Physical  condition  and  seasons  of  Saturn  . 


.  297 

299-304 

.  305 

.  306 


V.     Uranus  and  Neptune 

187.  Satellite  systems  of  Uranus  and  Neptune  . 

188.  Atmospheres  and  albedoes  of  Uranus  and  Neptune 

189.  Periods  of  rotation  of  Uranus  and  Neptune 

190.  Physical  conditions  of  Uranus  and  Neptune 


306 
307 
307 
308 


CHAPTER  X 
COMETS  AND   METEORS 


I.     Comets 

191.  General  appearance  of  comets   . 

192.  The  orbits  of  comets  . 

193.  194.   The  dimensions  and  masses  of  comets 
196.    Families  of  comets 

196.  The  capture  of  comets 

197.  On  the  origin  of  comets 

198.  Theories  of  comets'  tails 

199.  The  disintegration  of  comets     . 

200.  Historical  comets        .... 

201.  Halley's  comet 


.     311 

.     313, 

316,  317 

318 
.     320 

322 
.  323 
.     327 

328 
.     332 


II.     Meteors 


202.  Meteors,  or  "shooting  stars ' 

203.  The  number  of  meteors 

204.  205.   Meteoric  showers 


337 
338 
339 


CONTENTS  XV 


ARTS. 


'"""■  PAGE 

206.  Connection  between  comets  and  meteors  .        .        :        .        .  341 

207.  Effects  of  meteors  on  the  solar  system 34.3 

208.  Meteorites .        .  343 

Theories  respecting  the  origin  of  meteors 345 


209 


CHAPTER  XI 

THE  SUN 
I.     The  Sun's  Heat 

210.  The  problem  of  the  sun's  heat 349 

211.  Amount  of  energy  received  from  sun        ....  349 

212.  Sources  of  energy  used  by  man 351 

213.  Amount  of  energy  radiated  by  sun 353 

214.  The  temperature  of  the  sun  354 

215.  Principle  of  the  conservation  of  energy     .  ...  355 

216.  217.   Theories  of  the  sun's  heat 356-359 

218.  Fast  and  future  of  sun  on  contraction  theory    ....     360 

219.  The  age  of  the  earth 360 

w 

II.     Spectkum  Analysis 

220.  The  nature  of  light    ....  .  .365 

221.  On  the  production  of  light 366 

222.  Spectroscopes  and  the  spectrum 369 

223-226.   The  laws  of  spectrum  analysis 371-375 

III.     The  Constitution  of  the  Sun 

227.  Outline  of  the  sun's  constitution 378 

228.  The  photosphere 379 

229-231.    Sunspots,  distribution,  periodicity,  and  motions  .  381-384 

232.   The  rotation  of  the  sun 388 

238.   The  reversing  layer 390 

234.  Chemical  constitution  of  reversing  layer  .  392 

235,  236.   The  chromosphere  and  prominences  .  .        .     394,  395 

237.  The  spectroheliograph  398 

238.  The  corona 401 

239.  The  eleven-year  cycle ,        .        ,     404 


XVI 


CONTENTS 


CHAPTER  XII 
EVOLUTION  OF  THE   SOLAR  SYSTEM 
I.     General  Considerations  on  Evolution 

ARTS. 

240.  Essence  of  the  doctrine  of  evolution  .... 

241.  Value  of  a  theory  of  evolution 

242.  Outline  of  growth  of  doctrine  of  evolution 


PAOB 

407 
408 
410 


II.     Data  of  Frorlem  or  Evolution  of  Solar  System 

243.  General  evidences  of  orderly  development        .  .         .  413 

244.  Distribution  of  mass  in  the  solar  system  .  ■  .         .  414 

245.  Distribution  of  moment  of  momentum 416 

246.  The  energy  of  the  solar  system 419 


III.     The  Planetesimal  Theory 

247.  Outline  of  the  planetesimal  theory    . 

248.  Examples  of  planetesimal  organization 

249.  Suggested  origin  of  spiral  nebulse 

250.  The  origin  of  planets 

251.  The  planes  of  the  planetary  orbits     . 

252.  The  eccentricities  of  the  planetary  orbits 

253.  The  rotation  of  the  sun 

254.  The  rotation  of  the  planets 

255.  The  origin  of  satellites 

256.  The  rings  of  Saturn 

257.  258.   The  planetoids  and  zodiacal  light 

259.  The  comets         ... 

260.  The  future  of  the  solar  system  . 


421 
422 
424 
431 
433 
434 
436 
437 
440 
441 
442 
442 
443 


IV.     Historical  Cosmogonies 

261.  The  hypothesis  of  Kant     . 

262.  The  hypothesis  of  Laplace 

263.  264.   Tidal  forces  and  tidal  evolution 

265.  Effects  of  tides  on  motions  of  the  moon 

266.  Effects  of  tides  on  motions  of  the  earth 

267.  lldal  evolution  of  the  planets    . 


.  446 
.  449 
452,  454 
.  456 
.  456 
.     460 


CONTENTS 


XVil 


CHAPTER  XIII 
I.     The  Apparbjst  Distbibution  of  the  Stars 

ABT8.  piGE 

268.  On  the  problems  of  the  sidereal  universe 463 

269.  Number  of  stars  of  various  magnitudes      .                ...  464 

270.  Apparent  distribution  of  the  stars 470  ' 

271.  Form  and  structure  of  the  Milky  Way 473 


II.    Distances  and  Motions  op  the  Stars 

272.  Direct  parallaxes  of  nearest  stars 476 

273.  Distances  of  stars  from  proper  motions  and  radial  velocities    .  481 

274.  Motion  of  sun  with  respect  to  stars 482 

275.  Distances  of  stars  from  motion  of  sun 484 

276.  Kapteyn's  results  on  distances  of  stars 486 

277.  Distances  of  moving  groups  of  stars 487 

278.  Star  streams 490 

279.  On  the  dynamics  of  the  stellar  system 491 

280.  Runaway  stars 498 

281.  Globular  clusters 500 


III.     The  Stars 


282.  Double  stars 505 

283,  284.    Orbits  and  masses  of  binary  stars 507 

285,  286.    Spectroscopic  binary  stars 510 

287-293.   Variable  stars  of  various  types 515 

294.  Temporary  stars 523 

295.  The  spectra  of  the  stars  527 

296.  Phenomena  associated  with  spectral  types        ....  530 

297.  On  the  evolution  of  the  stars     .  ,  532 

298.  Tacit  assumptions  oif  theories  of  stellar  evolution     .        .        .  534 

299.  Origin  and  evolution  of  binary  stars 543 

300.  On  the  infinity  of  the  physio^  universe  in  space  and  in  time    .  548 


301. 
302. 
303. 
304. 


IV.     The  Nebdl^ 

Irregular  nebulse 550 

Spiral  nebulae 554 

Ring  nebulsB      .        .  "^^ 

Planetary  nebulae 560 


LIST   OF  TABLES 


HO. 

I.  The  first-magnitude  stars 

II.  Numbers  of  stars  in  first  six  magnitudes 

m.  The  constellations 

^       IV.  Elements  of  the  orbits  of  the  planets  . 

V.  Data  on  sun,  moon,  and  planets 

VI.  Dates  of  eastern  elongation  and  opposition 

VII.  Jupiter's  satellite  system     . 

VIII.  Saturn's  satellite  system 

IX.  Saturn's  ring  system    . 

X.  Rotation  of  the  sun  in  different  latitudes 

XI.  Elements  found  in  the  sun 

XII.  Distribution  of  moment  of  momentum  in  solar  system 

Xm.  Distances  of  ejection  for  various  initial  yelocities 

XIV.  Numbers  of  stars  in  magnitudes  5  to  17 

XV.  Distribution  of  the  stars  with  respect  to  the  Galaxy 

XVI.  Table  of  nineteen  nearest  stars   .... 

XVII.  Distances  of  stars  of  magnitudes  1  to  15 

XVIII.  Binary  stars  whose  masses  are  known 


PAGE 

144 
145 
147 
249 
254 
256 
290 
298 
300 
389 
393 
417 
428 
466 
471 
478 
486 
509 


LIST   OF  PHOTOGRAPHIC  ILLUSTRATIONS 


NO.  PAGE 

1.  The  Lick  Observatory,  Mt.  Hamilton,  Cal.       .        .       frontispiece 

2.  The  Yerkes  Observatory,  Williams  Bay,  Wis.  .        .       facing        1 

3.  The  moon  at  8.5  days  (Ritchey;  Yerkes  Observatory)     .        .      20 

24.  Orion  star  trails  (Barnard ;  Yerkes  Observatory)      .  .      77 

25.  Circumpolar  star  trails  (Ritchey) 78 

54.  The  40-inch  telescope  of  the  Yerkes  Observatory  .        .     138 

55.  The  Big  Dipper  (Hughes;  Yerkes  Observatory)        .         .  149 

57.  The  sickle  in  Leo  (Hughes;  Yerkes  Observatory)  .        .     157 

58.  Great  Andromeda  Nebula  (Ritchey  ;  Yerkes  Observatory)        .     158 

59.  The  Pleiades  (Wallace  ;  Yerkes  Observatory)  ....     161 

60.  Orion  ( Hughes  ;  Yerkes  Observatory) 163 

61.  Great  Orion  Nebula  (Ritchey  ;  Yerkes  Observatoi-y)  .  164 
68.  The  earth-lit  moon  (Barnard  ;  Yerkes  Observatory)  .  .  192 
75.  Moon  at  9|  days  (Ritchey  ;  Yerkes  Observatory)      .         .         .     208 

77.  The  Crater  Theophilus  (Ritchey  ;  Yerkes  Observatory)   .         .     210 

78.  Great  Crater  Clavius  (Ritchey ;  Yerkes  Observatory)  .     212 

79.  The  full  moon  (Wallace  ;  Yerkes  Observatory)  .         .215 

86.  Johann  Kepler  (Collection  of  David  Eugene  Smith)  \        .         .     229 

87.  Isaac  Newton  (Collection  of  David  Eugene  Smith)   .        .        .    232 

90.  William  Herschel  (Collection  of  David  Eugene  Smith)  .    239 

91.  John  Couch  Adams  (Collection  of  David  Eugene  Smith)  .    240 

92.  Joseph  Leverrier  (Collection  of  David  Eugene  Smith)  .  .  240 
99.  Trail  of  Planetoid  Egeria  (Parkhurst ;  Yerkes  Observatory)     .    259 

103.   Mars  (Barnard ;  Yerkes  Observatory) 275 

108.  Mars  (Mount  Wilson  Solar  Observatory)  .....    286 

113.  Jupiter  (E.  C.  Slipher ;  Lowell  Observatory)    .                .        .    295 

117.   Saturn  (Barnard  ;  Yerkes  Observatory) 301 

119.  Brooks'  Comet  (Barnard  ;  Yerkes  Observatory)       .        .        .    312 

124.  Delavan's  Comet  (Barnard ;  Yerkes  Observatory)    .        .        .    325 

125.  Encko's  Comet  (parnard ;  Yerkes  Observatory)  .  .  .  329 
1^6.  Morehouse's  Comet  (Barnard  ;  Yerkes  Observatory)  .  .  333 
128.  Halley's  Comet  (Barnard  ;  Yerkes  Observatory)      .        .        .335 

133.  Long  Island,  Kan.,  meteorite  (Farrington)       .        .        .        .344 

134.  Canon  Diablo,  Ariz.,  meteorite  (Farrington)    .        .        .        .346 


xxii       LIST   OF  PHOTOGRAPHIC  ILLUSTRATIONS 


PAGE 


135.  Durango,  Mexico,  meteorite  (Farrington)         .        .        .        .345 

136.  Tower  telescope  of  the  Mt.  Wilson  Solar  Observatory       .        .  348 

141.   Tlie  sun  (Fox ;  Yerkes  Observatory) 376 

144.   Sun  spot,  July  17,  1905  (Fox ;  Yerkes  Observatory)                 .  382 

146.  Sunspots  with  opposite  polarities  (Hale;  Solar  Observatory)  .  386 

147.  SolarObservatory  of  the  Carnegie  Institution,  Mt.  Wilson,  Cal.  387 

149.  Solar  prominence  80,000  miles  high  (Solar  Observatory)  .         .  396 

150.  Motion  in  solar  prominences  (Slocum  ;  Yerkes  Observatory)  .  397 

152.  Spectroheliogram  of  sun  (Hale  and  EUerman;  Yerkes  Observa- 

tory)   .                .                .        .                                 .        .  400 

153.  Spectroheliograms  of  sun  spot  (Hale  and  EUerman ;  Solar  Ob- 

servatory)  .                                 ...                .        .  401 

154.  The  sun's  corona  (Barnard  and  Ritchey)           ....  402 
157.   Eruptive  prominences  (Slocum ;  Yerkes  Observatory)      .         .  426 

159.  Great  spiral  nebula  M.  51  (Ritchey ;  Yerkes  Observatory)        .  429 

160.  Great  spiral  nebula  M.  33  (Ritchey  ;  Yerkes  Observatory)  430 
162.  Laplace  (Collection  of  David  Eugene  Smith)     .        .        .  449 

165.  Milky  Way  in  Aquila  (Barnard ;  Yerkes  Observatory) 

166.  Star  clouds  in  Sagittarius  (Barnard  ;  Yerkes  Observatory) 

167.  Region  of  Rho  Ophiuchi  (Barnard  ;  Yerkes  Observatory) 
171.   Hercules  star  cluster  (Ritchey  ;  Yerkes  Observatory) 

173.  Spectra  of  Mizar  (Frost ;  Yerkes  Observatory) 

174.  Spectra  of  Mu  Orionis  (Frost ;  Yerkes  Observatory) 

180.  Nova  Persei  (Ritchey ;  Yerkes  Observatory)     . 

181.  The  spectrum  of  Sirius  (Yerkes  Observatory)  . 

182.  The  spectrum  of  Beta  Geminorum  (Yerkes  Observatory) 

183.  The  spectrum  of  Arcturus  (Yerkes  Observatory)      .         .  529 

184.  The  Pleiades  (Ritchey ;  Yerkes  Observatory)  .        .  537 

187.  Nebula  in  Cygnus  (Ritchey  ;  Yerkes  Observatory)   .  651 

188.  Bright  and  dark  nebulae  (Barnard  ;  Yerkes  Observatory)  554 

189.  The  Trifid  Nebula  (Crossley  reflector  ;  Lick  Observatory)  555 

190.  Spiral  nebula  in  Ursa  Major  (Ritchey  ;  Yerkes  Observatory)  .  556 

191.  Spiral  nebula  in  Andromeda  (Crossley  reflector ;  Lick  Observ- 

atory)           657 

192.  Great  nebula  in  Andromeda  (Ritchey  ;  Yerkes  Observatory)   .  559 

193.  Ring  nebula  in  Lyra  (Sullivan  ;  Yerkes_Observatory)       .         .  660 

194.  Planetary  nebula  (Crossley  reflector  ;  Lick  Observatory)          .  561 


462 
472 
474 
501 
511 
513 
525 
527 
528 


AN  INTKODUCTION  TO  ASTEONOMY 


A:N^  mTRODUCTIOJN^  TO 
ASTROIN^OMY 

CHAPTER  I 
PRELIMINARY    CONSIDERATIONS 

1.  Science.  —  The  progress  of  mankind  has  been  marked 
by  a  number  of  great  intellectual  movements.  At  one  tinie 
the  ideas  of  men  were  expanding  with  the  knowledge  which 
they  were  obtaining  from  the  voyages  of  Columbus,  Magel- 
lan, and  the  long  list  of  hardy  explorers  who  first  visited  the 
remote  parts  of  the  earth.  At  another,  milUons  of  men  laid 
down  their  lives  in  order  that  they  might  obtain  toleration 
in  religious  behefs.  At  another,  the  struggle  was  for  poUtical 
freedom.  It  is  to  be  noted  with  satisfaction  that  those 
movements  which  have  involved  the  great  mass  of  people, 
from  the  highest  to  the  lowest,  have  led  to  results  which 
have  not  been  lost. 

The  present  age  is  known  as  the  age  of  science.  Never 
before  have  so  many  men  been  actively  engaged  in  the 
pursuit  of  science,  and  never  before  have  its  results  con- 
tributed so  enormously  to  the  ordinary  affairs  of  life.  If  all 
its  present-day  appUcations  were  suddenly  and  for  a  con- 
siderable time  removed,  the  results  would  be  disastrous. 
With  the  stopping  of  trains  and  steamboats  the  food  supply 
in  cities  would  soon  fail,  and  there  would  be  no  fuel  with 
which  to  heat  the  buildings.  Water  could  no  longer  be 
pumped,  and  devastating  fires  might  follow.  If  people  es- 
caped to  the  country,  they  would  perish  in  large  numbers 


2  AN  INTRODUCTION  TO  ASTRONOMY      [ch.  i,  1 

because  without  modern  machinery  not  enough  food  could 
be  raised  to  supply  the  population.  In  fact,  the  more  the 
subject  is  considered,  the  more  clearly  it  is  seen  that  at  the 
present  time  the  lives  of  civihzed  men  are  in  a  thousand  ways 
directly  dependent  on  the  things  produced  by  science. 

Astronomy  is  a  science.  That  is,  it  is  one  of  those  sub- 
jects, such  as  physics,  chemistry,  geology,  and  biology, 
which  have  made  the  present  age  in  very  many  respects 
altogether  different  from  any  earlier  one.  Indeed,  it  is  the 
oldest  science  and  the  parent  of  a  number  of  the  others,  and, 
in  many  respects,  it  is  the  most  perfect  one.  For  these  rea- 
sons it  illustrates  most  simply  and  clearly  the  characteristics 
of  science.  Consequently,  when  one  enters  on  the  study 
of  astronomy  he  not  only  begins  an  acquaintance  with  a 
subject  which  has  always  been  noted  for  its  lofty  and  un- 
selfish ideals,  but,  at  the  same  time,  he  becomes  famihar  with 
the  characteristics  of  the  scientific  movement. 

2.  The  Value  of  Science.  —  The  importance  of  science 
in  changing  the  relations  of  men  to  the  physical  universe 
about  them  is  easy  to  discern  and  is  generally  more  or  less 
recognized.  That  the  present  conditions  of  life  are  better 
than  those  which  prevailed  in  earlier  times  proves  the  value 
of  science,  and  the  more  it  is  considered  from  this  point  of 
view,  the  more  valuable  it  is  found  to  be. 

The  changes  in  the  mode  of  Hving  of  man  which  science 
has  brought  about,  will  probably  in  the  course  of  time  give 
rise  to  marked  alterations  in  his  physique;  for,  the  better 
food  supply,  shelter,  clothing,  and  sanitation  which  have 
recently  been  introduced  as  a  consequence  of  scientific  dis- 
coveries, correspond  in  a  measure  to  the  means  by  which  the 
best  breeds  of  domestic  animals  have  been  developed,  and 
without  which  they  degenerate  toward  the  wild  stock  from 
which  they  have  been  derived.  And  probably,  also,  as  the 
factors  which  cause  changes  in  living  organisms  become 
better  known  through  scientific  investigations,  man  will 
consciously  direct  his  own  evolution. 


CH.  I,  2]         PRELIMINARY  CONSIDERATIONS  3 

But  there  is  another  less  speculative  respect  in  which 
science  is  important  and  in  which  its  importance  will  enor- 
mously increase.  It  has  a  profound  influence  on  the  minds 
of  those  who  devote  themselves  to  it,  and  the  number  of 
those  who  are  interested  in  it  is  rapidly  increasing.  In  the 
first  place,  it  exalts  truth  and  honestly  seeks  it,  wherever  the 
search  may  lead.  In  the  second  place,  its  subject  matter 
often  gives  a  breadth  of  vision  which  is  not  otherwise  ob- 
tained. For  example,  the  complexity  and  adaptability  of 
living  beings,  the  irresistible  forces  which  elevate  the  moun- 
tains, or  the  majestic  motions  of  the  stars  open  an  intellectual 
horizon  far  beyond  that  which  belongs  to  the  ordinary  af- 
fairs of  life.  The  conscious  and  deUberate  search  for  truth 
and  the  contemplation  of  the  wonders  of  nattire  change  the 
mental  habits  of  a  man.  They  tend  to  make  him  honest 
with  himself,  just  in  his  judgment,  and  serene  in  the  midst 
of  petty  annoyances.  In  short,  the  study  of  science  makes 
character,  as  is  splendidly  illustrated  in  the  hves  of  many 
celebrated  scientific  men.  It  would  undoubtedly  be  of  very 
great  benefit  to  the  world  if  every  one  could  have  the  dis- 
cipUne  of  the  sincere  and  honest  search  for  the  truth  which 
is  given  by  scientific  study,  and  the  broadening  influence  of 
an  acquaintance  with  scientific  theories. 

There  is  an  important  possible  indirect  effect  of  science 
on  the  intellectual  development  of  mankind  which  should 
not  be  overlooked.  One  of  the  results  of  scientific  discoveries 
has  been  the  greatly  increased  productivity  of  the.  human 
race.  All  of  the  necessities  of  life  and  many  of  its  luxuries 
can  now  be  supplied  by  the  expenditure  of  much  less  time 
than  was  formerly  required  to  produce  the  bare  means  of 
existence.  This  leaves  more  leisure  for  intellectual  pm-suits. 
Aside  from  its  direct  effects,  this  is,  when  considered  in  its 
broad  aspects,  the  most  important  benefit  conferred  by 
science,  because,  in  the  final  analysis,  intellectual  experiences 
are  the  only  things  in  which  men  have  an  interest.  As  an 
illustration,  any  one  would  prefer  a  normal  conscious  life 


4      AN  INTRODUCTION  TO  ASTRONOMY  [ch.  i,  2 

for  one  year  rather  than  an  existence  of  five  hundred  years 
with  the  certainty  that  he  would  be  completely  unconscious 
during  the  whole  time. 

It  is  often  supposed  that  science  and  the  fine  arts,  whose 
importance  every  one  recognizes,  are  the  antitheses  of  each 
other.  The  arts  are  beUeved  to  be  warm  and  human,  — 
science,  cold  and  austere.  This  is  very  far  from  being  the 
case.  While  science  is  exacting  in  its  demands  for  pre- 
cision, it  is  not  insensible  to  the  beauties  of  its  subject.  In 
all  branches  of  science  there  are  wonderful  harmonies  which 
appeal  strongly  to  those  who  fully  comprehend  them.  Many 
of  the  great  scientists  have  expressed  themselves  in  their 
writings  as  being  deeply  moved  by  the  aesthetic  side  of  their 
subject.  Many  of  then!  have  had  more  than  ordinary  taste 
for  art.  Mathematicians  are  noted  for  being  gifted  in  music, 
and  there  are  numerous  examples  of  scientific  men  who 
were  fond  of  painting,  sculpture,  or  poetry.  But  even  if 
the  common  opinion  that  science  and  art  are  opposites  were 
correct,  yet  science  would  contribute  indirectly  to  art  through 
the  leisure  it  furnishes  men. 

3.  The  Origin  of  Science.  —  It  is  doubtful  if  any  impor- 
tant scientific  idea  ever  sprang  suddenly  into  the  mind  of  a 
single  man.  The  great  intellectual  movements  in  the  world 
have  had  long  periods  of  preparation,  and  often  many  men 
were  groping  for  the  same  truth,  without  exactly  seizing  it, 
before  it  was  fully  eonapreh ended. 

The  foundation  on  which  all  science  rests  is  the  principle 
that  the  imiverse  is  orderly,  and  that  all  phenomena  succeed 
one  another  in  harmony  with  invariable  laws.  Consequently, 
science  was  impossible  until  the  truth  of  this  principle  was 
perceived,  at  least  as  appUed  to  a  hmited  part  of  nature. 

The  phenomena  of  ordinary  observation,  as,  for  example, 
the  weather,  depend  on  such  a  multitude  of  factors  that  it 
was  not  easy  for  men  in  their  primitive  state  to  discover 
that  they  occur  in  harmony  with  fixed  laws.  This  was  the 
age  of  superstition,  when  nature  was  supposed  to  be  con- 


CH.  I,  3]        PRELIMINARY  CONSIDERATIONS  5 

trolled  by  a  great  number  of  capricious  gods  whose  favor 
could  be  won  by  childish  ceremonies.  Enormous  experience 
was  required  to  dispel  such  errors  and  to  convince  men  that 
the  universe  is  one  vast  organization  whose  changes  take 
place  in  conformity  with  laws  which  they  can  in  no  way 
•alter. 

The  actual  dawn  of  science  was  in  prejiistorie  times, 
probably  in  the  civilizations  that  flourished  in  the  valleys 
of  the  Nile  and  the  Euphrates.  In  the  very  earliest  records 
of  these  people  that  have  come  down  to  modern  times  it  is 
foimd  that  they  were  acquainted  with  many  astronomical 
phenomena  and  had  coherent  ideas  with  respect  to  the  mo- 
tions of  the  sun,  moon,  planets,  and  stars.  It  is  perfectly 
clear  from  their  writings  that  it  was  from  their  observations 
of  the  heavenly  bodies  that  they  first  obtained  the  idea  that 
the  universe  is  not  a  chaos.  Day  and  night  were  seen  to 
succeed  each  other  regularly,  the  moon  was  found  to  pass 
through  its  phases  systematically,  the  seasons  followed  one 
another  in  order,  and  in  fact  the  more  conspicuous  celestial 
phenomena  were  observed  to  occur  in  an  orderly  sequence. 
It  is  to  the  glory  of  astronomy  that  it  first  led  men  to  the 
conclusion  that  law  reigns  in  the  universe. 

The  ancient  Greeks,  at  a  period  four  or  five  hundred 
years  preceding  the  Christian  era,  definitely  undertook  to 
find  from  systematic  observation  how  celestial  phenomena 
follow  one  another.  They  determined  very  accurately  the 
number  of  days  in  the  year,  the  period  of  the  moon's  revolu- 
tion, and  the  paths  of  the  sun  and  the  moon  among  the 
stars ;  they  correctly  explained  the  cause'  of  eclipses  and 
learned  how  to  predict  them  with  a  considerable  degree  of 
accuracy;  they  undertook  to  measure  the  distances  to  the 
heavenly  bodies,  and  to  work  out  a  complete  system  that 
would  represent  their  motions.  The  idea  was  current  among 
the  Greek  philosophers  that  the  earth  was  spherical,  that  it 
turned  on  its  axis,  and,  among  some  of  them,  that  it  revolved 
around  the  sun.    They  had  true  science  in  the  modem 


6  AN  INTRODUCTION  TO  ASTRONOMY     [ch.  i,  3 

acceptance  of  the  term,  but  it  was  largely  confined  to  the 
relations  among  celestial  phenomena.  The  conception  that 
the  heavens  are  orderly,  which  they  definitely  formulated  and 
acted  on  with  remarkable  success,  has  been  extended,  espe- 
cially in  the  last  two  centuries,  so  as  to  include  the  whole 
universe.  The  extension  was  first  made  to  the  inanimate 
world  and  then  to  the  more  comphcated  phenomena  asso- 
ciated with  Uving  beings.  Every  increase  in  carefully 
recorded  experience  has  confirmed  and  strengthened  the 
belief  that  nature  is  perfectly  orderly,  until  now  every  one 
who  has  had  an  opportunity  of  becoming  familiar  with  any 
science  is  firmly  convinced  of  the  truth  of  this  principle, 
which  is  the  basis  of  all  science. 

4.  The  Methods  of  Science.  —  Science  is  concerned  with 
the  relations  among  jfcenomena,  and  it  must  therefore  rest 
ultimately  upon  observations  and  experiments.  Since  its 
ideal  is  exactness,  the  observations  and  experiments  must 
be  made  with  all  possible  precision  and  the  results  must  be 
carefully  recorded.  These  principles  seem  perfectly  obvious, 
yet  the  world  has  often  ignored  them.  One  of  the  chief 
faults  of  the  scientists  of  ancient  times  was  that  th^y  indulged 
in  too  many  arguments,  more  or  less  metaphysical  in  charac- 
ter, and  made  too  few  appeals  to  what  would  now  seem  ob- 
vious observation  or  experiment.  A  great  Enghsh  philoso- 
pher, Roger  Bacon  (1214-1294),  made  a  powerful  argument 
in  favor  of  founding  science  and  philosophy  on  experience. 

It  must  not  be  supposed  that  the  failure  to  rely  on  obser- 
vations and  experiment,  and  especially  to  record  the  results 
of  experience,  are  faults  that  the  world  has  outgrown.  On 
the  contrary,  they  are  still  almost  imiversally  prevalent 
among  men.  •  For  example,  there  are  many  persons  who  be- 
lieve in  dreams  or  premonitions  because  once  in  a  thousand 
cases  a  dream  or  a  premonition  comes  true.  If  they  had 
written  down  in  every  case  what  was  expected  and  what 
actually  happened,  the  absurdity  of  their  theory  would 
have  been  evident.     The  whole  mass  of  superstitions  with 


CH.  I,  4]        PRELIMINARY  CONSIDERATIONS  7 

which  mankind  has  burdened  itself  survives  only  because 
the  results  of  actual  experience  are  ignored. 

In  scientific  work  great  precision  in  making  observations 
and  experiments  is  generally  of  the  highest  importance. 
Every  science  furnishes  examples  of  cases  where  the  data 
seemed  to  have  been  obtained  with  greater  exactness  than 
was  really  necessary,  and  where  later  the  extra  accuracy 
led  to  important  discoveries.  In  this  way  the  foundation 
of  the  theory  of  the  motion  of  the  planets  was  laid.  Tycho 
Brahe  was  an  observer  not  only  of  extraordinary  industry, 
but  one  who  did  all  his  work  with  the  most  painstaking  care. 
Kepler,  who  had  been  his  pupil  and  knew  of  the  excellence 
of  his  measurements,  was  a  computer  who  sought  to  bring 
theory  and  observation  into  exact  harmony.  He  foimd  it 
impossible  by  means  of  the  epicycles  and  eccentrics,  which 
his  predecessors  had  used,  to  represent  exactly  the  observa- 
tion of  Tycho  Brahe.  In  spite  of  the  fact  that  the  discrep- 
ancies were  small  and  might  easily  have  been  ascribed  to 
errors  of  observation,  Kepler  had  absolute  confidence  in 
his  master,  and  by  repeated  trials  and  an  enormous  amount 
of  labor  h'i  finally  arrived  at  the  true  laws  of  planetary 
motion  (Art.  145).  These  laws,  in  the  hands  of  the  genius 
Newton,  led  directly  to  the  law  of  gravitation  and  to  the 
explanation  of  all  the  many  peculiarities  of  the  motions  of 
the  moon  and  planets  (Art.  146) . 

Observations  alone,  however  carefully  they  may  have 
been  made  and  recorded,  do  not  constitute  science.  First, 
the  phenomena;  must  be  related,  and  then,  what  they  have 
in  common  must  be  perceived.  It  might  seem  that  it  would 
be  a  simple  matter  to  note  in  what  respects  phenomena  are 
similar,  but  experience  has  shown  that  only  a  very  few  have 
the  ability  to  discover  relations  that  are  not  already  known. 
If  this  were  not  true,  there  would  not  be  so  many  examples 
of  new  inventions  and  discoveries  depending  on  very  simple 
things  that  have  long  been  within  the  range  of  experience  of 
every  one.     After  the  common  element  in  the  observed 


8  AN  INTRODUCTION  TO  ASTRONOMY      [ch.  i,  4 

phenomena  has  been  discovered  the  next  step  is  to  infer,  by 
the  process  known  as  induction,  that  the  same  thing  is  true 
in  all  similar  cases.  Then  comes  the  most  difficult  thing  of 
all.  The  vital  relationships  of  the  one  class  of  phenomena 
with  other  classes  of  phenomena  must  be  discovered,  and 
the  several  classes  must  be  organized  into  a  coherent  whole. 

An  illustration  will  make  the  process  clearer  than  an 
extended  argument.  Obviously,  all  men  have  observed 
moving  bodies  all  their  Uves,  yet  the  fact  that  a  moving  body, 
subject  to  no  exterior  force,  proceeds  in  a  straight  Une  with 
uniform  speed  was  not  known  until  about  the  time  of  GaHleo 
(1564-1642)  and  Newton  (1643-1727).  When  the  result 
is  once  enunciated  it  is  easy  to  recall  many  confirmatory 
experiences,  and  it  now  seems  remarkable  that  so  simple 
a  fact  should  have  remained  so  long  undiscovered.  It  was 
also  noted  by  Newton  that  when  a  body  is  acted  on  by  a 
force  it  has  an  acceleration  (acceleration  is  the  rate  of 
change  of  velocity)  in  the  direction  in  which  the  force  acts, 
and  that  the  acceleration  is  proportional  to  the  magnitude 
of  the  force.  Dense  bodies  left  free  in  the  air  fall  toward 
the  earth  with  accelerated  velocity,  and  they  are  therefore 
subject  to  a  force  toward  the  earth.  Newton  observed  these 
things  in  a  large  number  of  cases,  and  he  inferred  by  induc- 
tion that  they  are  universally  true.  He  focused  particularly 
on  the  fact  that  every  body  is  subject  to  a  force  directed 
toward  the  earth. 

If  taken  alone,  the  fact  that  bodies  are  subject  to  forces 
toward  the  earth  is  not  so  very  important;  but  Newton 
used  it  in  cormection  with  many  other  phenomena.  For 
example,  he  knew  that  the  moon  is  revolving  around  the 
earth  in  an  approximately  circular  orbit.  At  P,  in  Fig.  3, 
it  is  moving  in  the  direction  PQ  around  the  earth,  E.  But 
it  actually  moves  from  P  to  R.  That  is,  it  has  fallen  toward 
the  earth  through  the  distance  QR.  Newton  perceived  that 
this  motion  is  analogous  to  that  of  a  body  falling  near  the 
surface  of  the  earth,  or  rather  to  the  motion  of  a  body  which 


CH.  r,  4]        PRELIMINARY  CONSIDERATIONS  9 

has  been  started  in  a  horizontal  direction  from  p  near  the 
surface  of  the  earth.  For,  if  the  body  were  started  hori- 
zontally, it  would  continue  in  the  straight  line  pg,  instead  of 
curving  downward  to  r,  if  it  were  not  acted  upon  by  a  force 
directed  toward  the  earth.  Newton  also 
knew  Kepler's  laws  of  planetary  motion. 
By  combining  with  wonderful  insight  a 
number  of  classes  of  phenomena  which 
before  his  time  had  been  supposed  to  be 
unrelated,  he  finally  arrived  at  the  law  of 
gravitation — "  Every  particle  of  matter 
in  the  universe  attracts  every  other  par- 
ticle with  a  force  which  is  directly  pro- 
portional to  the  product  of  their  masses 
and  inversely  proportional  to  the  square 
of  their  distance  apart."  Thus,  by  per- 
ceiving the  essentials  in  many  kinds  of  Fig.  3.  —  The  motion 
phenomena  and  by  an  almost  unparal-     ?*  ^^  moon  from  p 

,  Z     .  ,  .    .  ,  to "  around  E  is  smu- 

leled  stroke  or  gemus  m  combmmg  them,     lar  to  that  of  a  body 
he  discovered  one  of  the  relations  which     Projected  horizontally 

from  p. 

every  particle  of  matter  in  the  universe 
has  to  all  the  others.      By  means  of  the  laws  of  motion 
(Art.  40)  and  the  law  of  gravitation,  the  whole  problem  of 
the  motions  of  bodies  was  systematized. 

There  is  still  another  method  employed  in  science  which 
is  often  very  important.  After  general  principles  have 
been  discovered  they  can  be  used  as  the  basis  for  deducing 
particular  conclusions.  The  value  of  the  particular  conclu- 
sions may  consist  in  leading  to  the  accomplishment  of  some 
desired  end.  For  example,  since  a  moving  body  tends  to 
continue  in  a  straight  line,  those  who  build  railways  place 
the  outside  rails  on  curves  higher  than  those  on  the  in- 
side so  that  trains  will  not  leave  the  track.  Or,  the 
knowledge  of  the  laws  of  projectiles  enables  gunners  to  hit 
invisible  objects  whose  positions  are  known. 

The  value  of  particular  conclusions  may  consist  in  ena- 


10  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  i,  4 

bling  men  to  adjust  themselves  to  phenomena  over  which 
they  have  no  control.  For  example,  in  many  harbors  large 
boats  can  enter  or  depart  only  when  the  tide  is  high,  and  the 
knowledge  of  the  times  when  the  tides  will  be  high  is  valuable 
to  navigators.  After  the  laws  of  meteorology  have  become 
more  perfectly  known,  so  that  approaching  storms,  or 
frosts,  or  drouths,  or  hot  waves  can  be  accurately  foretold 
a  considerable  time  in  advance,  the  present  enormous  losses 
due  to  these  causes  will  be  avoided. 

The  knowledge  of  general  laws  may  lead  to  information 
regarding  things  which  are  altogether  inaccessible  to  obser- 
vation or  experiment.  For  example,  it  is  very  important 
for  the  geologist  to  know  whether  the  interior  of  the  earth 
is  solid  or  liquid ;  and,  if  it  is  solid,  whether  it  is  elastic  or 
viscous.  Although  at  first  thought  it  seems  impossible  to 
obtain  reliable  information  on  this  subject,  yet  by  a  number 
of  indirect  processes  (Arts.  25,  26)  based  on  laws  established 
from  observation,  it  has  been  possible  to  prove  with  cer- 
tainty that  the  earth,  through  and  through,  is  about  as 
rigid  as  steel,  and  that  it  is  highly  elastic. 

Another  important  use  of  the  deductive  process  in  science 
is  in  drawing  consequences  of  a  theory  which  must  be  ful- 
filled in  experience  if  the  theory  is  correct,  and  which  may 
fail  if  it  is  false.  It  is,  indeed,  the  most  efiicient  means  of 
testing  a  theory.  Some  of  the  most  noteworthy  examples 
of  its  application  have  been  in  connection  with  the  law  of 
gravitation.  Time  after  time  mathematicians,  using  this 
law  as  a  basis  for  their  deductions,  have  predicted  phenom- 
ena that  had  not  been  observed,  and  time  after  time  their 
predictions  have  been  fulfilled.  This  is  one  of  the  reasons 
why  the  truth  of  the  law  of  gravitation  is  regarded  as  having 
been  firmly  estabUshed. 

5.  The  Imperfections  of  Science.  —  One  of  the  char- 
acteristics of  science  is  its  perfect  candor  and  fairness.  It 
would  not  be  in  harmony  with  its  spirit  to  attempt  to  lead 
one  to  suppose  that  it  does  not  have  sources  of  weakness. 


CH.  I,  5]        PRELIMINARY  CONSIDERATIONS  11 

Besides,  if  its  possible  imperfections  are  analyzed,  they  can 
be  more  easily  avoided,  and  the  real  nature  of  the  final 
conclusions  will  be  better  understood. 

It  must  be  observed,  in  the  first  place,  that  science  con- 
sists of  men's  theories  regarding  what  is  true  in  the  universe 
about  them.  These  theories  are  based  on  observation  and 
experiment  and  are  subject  to  the  errors  and  incompleteness 
of  the  data  on  which  they  are  founded.  The  fact  that  it 
is  not  easy  to  record  exactly  what  one  may  have  attempted 
to  observe  is  illustrated  by  the  divergence  in  the  accounts 
of  different  witnesses  of  anything  except  the  most  trivial 
occurrence.  Since  men  are  far  from  being  perfect,  errors 
in  the  observations  cannot  be  entirely  avoided,  but  in  good 
science  every  possible  means  is  taken  for  ehminating  them. 

In  addition  to  this  source  of  error,  there  is  another  more 
insidious  one  that  depends  upon  the  fact  that  observational 
data  are  often  collected  for  the  purpose  of  testing  a  specific 
theory.  If  the  theory  in  question  is  due  to  the  one  who  is 
making  the  observations  or  experiments,  it  is  especially 
diflBcult  for  him  to  secure  data  uninfluenced  by  his  bias  in 
its  favor.  And  even  if  the  observer  is  not  the  author  of  the 
theory  to  which  the  observations  relate,  he  is  very  apt  to  be 
prejudiced  either  in  its  favor  or  against  it. 

Even  if  the  data  on  which  science  is  based  were  always 
correct,  they  would  not  be  absolutely  exhaustive,  and  the 
inductions  to  general  principles  from  them  would  be  sub- 
ject to  corresponding  uncertainties.  Similarly,  the  general 
principles,  derived  from  various  classes  of  phenomena,  which 
are  used  in  formulating  a  complete  scientific  theory,  do  not 
include  all  the  principles  which  are  involved  in  the  particular 
domain  of  the  theory.  Consequently  it  may  be  imperfect 
for  this  reason  also. 

The  sources  of  error  in  scientific  theories  which  have  been 
enumerated  are  fundamental  and  will  always  exist.  The 
best  that  can  be  done  is  to  recognize  their  existence  and  to 
minimize  their  effects  by  all  possible  means.    The  fact  that 


12  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  i,  5 

science  is  subject  to  imperfections  does  not  mean  that  it  is 
of  little  value  or  that  less  effort  should  be  put  forth  in  its 
cultivation.  Wood  and  stone  and  brick  and  glass  have 
never  been  made  into  a  perfect  house ;  yet  houses  have  been 
very  useful  and  men  will  continue  to  build  them. 

There  are  many  examples  of  scientific  theories  which  it 
has  been  found  necessary  to  modify  or  even  to  abandon. 
These  changes  have  not  been  more  numerous  than  they 
have  been  in  other  domains  of  hmnan  activities,  but  they 
have  been,  perhaps,  more  frankly  confessed.  Indeed,  there 
are  plenty  of  examples  where  scientists  have  taken  evident 
satisfaction  in  the  alterations  they  have  introduced.  The 
fact  that  scientific  theories  have  often  been  found  to  be 
imperfect  and  occasionally  positively  wrong,  have  led  some 
persons  who  have  not  given  the  question  serious  consideration 
to  suppose  that  the  conclusions  of  science  are  worthy  of  no 
particular  respect,  and  that,  in  spite  of  the  pretensions  of 
scientists,  they  are  actually  not  far  removed  from  the  level 
of  superstitions.  The  respect  which  scientific  theories 
deserve  and  the  gulf  that  separates  them  from  superstitions 
will  be  evident  from  a  statement  of  their  real  nature. 

Suppose  a  person  were  so  situated  that  he  could  look 
out  from  an  upper  window  over  a  garden.  He  could  make 
a  drawing  of  what  he  saw  that  would  show  exactly  the  relative 
positions  of  the  walks,  shrubs,  and  flowers.  If  he  were  color 
bUnd,  the  drawing  could  be  made  in  pencil  so  as  to  satisfy 
perfectly  all  his  observations.  But  suppose  some  one  else 
who  was  not  color  blind  should  examine  the  drawing.  He 
would  legitimately  complain  that  it  was  not  correct  because 
the  colors  were  not  shown.  If  the  colors  were  correctly  given, 
both  observers  would  be  completely  satisfied.  Now  suppose 
a  third  person  should  look  at  the  drawing  and  should  then 
go  down  and  examine  the  garden  in  detail.  He  would  find 
that  the  various  objects  in  it  not  only  have  positions  but 
also  various  heights.  He  would  at  once  note  that  the 
heights  were  not  represented  in  the  drawing,  and  a  little 


CH.  I,  5]        PRELIMINARY  CONSIDERATIONS  13 

reflection  would  convince  him  that  the  three-dimensional 
garden  could  not  be  completely  represented  in  a  two-dimen- 
sional drawing.  He  would  claim  that  that  method  of  trying 
to  give  a  correct  idea  of  what  was  in  the  garden  was  funda- 
mentally wrong,  and  he  might  suggest  a  model  of  suitable 
material  in  three  dimensions.  Suppose  the  three-dimen- 
sional model  were  made  satisfying  the  third  observer.  It  is 
important  to  note  that  it  would  correctly  represent  all  the 
relative  positions  observed  by  the  first  one  and  all  the  colors 
observed  by  the  second  one,  as  well  as  the  additional  in- 
formation obtained  by  the  third  one. 

A  scientific  theory  is  founded  on  the  work  of  one  or  more 
persons  having  only  hmited  opportunities  for  observation 
and  experiment.  It  is  a  picture  in  the  imagination,  not  on 
paper,  of  the  portion  of  the  universe  under  consideration.  It 
represents  all  the  observed  relations,  and  it  is  assumed  that 
it  will  represent  the  relations  that  might  be  observed  in 
all  similar  circumstances.  Suppose  some  new  facts  are 
discovered  which  are  not  covered  by  the  theory,  just  as  the 
second  observer  in  the  garden  saw  colors  not  seen  by  the 
first.  It  will  be  necessary  to  change  the  scientific  theory  so 
as  to  include  them.  Perhaps  it  can  be  done  simply  by  adding 
to  the  theory.  But  if  the  new  facts  correspond  to  the  things 
discovered  by  the  third  observer  in  the  garden,  it  will  be 
necessary  to  abandon  the  old  theory  and  to  construct  an 
entirely  new  one.  The  new  one  must  preserve  all  the  rela- 
tions represented  by  the  old  one,  and  it  must  represent  the 
new  ones  as  well. 

In  the  light  of  this  discussion  it  may  be  asked  in  what 
sense  scientific  theories  are  true.  The  answer  is  that  they 
are  all  true  to  the  extent  that  they  picture  nature.  The 
relations  are  the  important  things.  When  firmly  established 
they  are  a  permanent  acquisition;  however  the  mode  of 
representing  them  may  change,  they  remain.  A  scientific 
theory  is  a  convenient  and  very  useful  way  of  describing  the 
relations  on  which  it  is  based.     It   correctly  represents 


14  AN   INTRODUCTION  TO  ASTRONOMY     [ch.  i,  5 

them,  and  in  this  respect  differs  from  a  superstition  which 
is  not  completely  in  harmony  with  its  own  data.  It  implies 
many  additional  things  and  leads  to  their  investigation.  If 
the  impUcations  are  found  to  hold  true  in  experience,  the 
theory  is  strengthened ;  if  not,  it  must  be  modified.  Hence, 
there  should  be  no  reproach  in  the  fact  that  a  scientific 
theory  must  be  altered  or  abandoned.  The  necessity  for 
such  a  procedure  means  that  new  information  has  been 
obtained,  not  that  the  old  was  false.' 

6.  Great  Contributions  of  Astronomy  to  Science.  —  As 
was  explained  in  Art.  3,  science  started  in  astronomy.  Many 
astronomical  phenomena  are  so  simple  that  it  was  possible 
for  primitive  people  to  get  the  idea  from  observing  them 
that  the  universe  is  orderly  and  that  they  could  discover  its 
laws.  In  other  sciences  there  are  so  many  varying  factors 
that  the  uniformity  in  a  succession  of  events  would  not  be 
discovered  by  those  who  were  not  deliberately  looking  for  it. 
It  is  sufficient  to  consider  the  excessive  complexities  of  the 
weather  or  of  the  developments  of  plants  or  animals,  to  see 
how  hopeless  would  be  the  problem  which  a  people  with- 
out a  start  on  science  would  face  if  they  were  cut  off  from 
celestial  phenomena.  It  is  certain  that  if  the  sky  had  al- 
ways been  covered  by  clouds  so  that  men  could  not  have 
observed  the  regular  motions  of  the  sun,  moon,  and  stars, 
the  dawn  of  science  would  have  been  very  much  delayed. 
It  is  entirely  possible,  if  not  probable,  that  without  the  help 
of  astronomy  the  science  of  the  human  race  would  yet  be  in 
a  very  primitive  state. 

Astronomy  has  made  positive  and  important  contribu- 
tions to  science  within  historical  times.  Spherical  trigo- 
nometry was  invented  and  developed  because  of  its  uses  in 
determining  the  relations  among  the  stars  on  the  vault  of 
the  heavens.     Very  many  things  in  calculus  and  still  higher 

'  The  comparison  of  scientific  theories  with  the  picture  of  the  objecta 
seen  in  the  garden  is  for  the  purpose  of  making  clear  one  of  their  particular 
features.  It  must  be  remembered  that  in  most  respects  the  comparison 
with  so  trivial  a  thing  is  very  imperfect  and  unfair  to  science. 


_CH.  I,  6]        PRELIMINARY  CONSIDERATIONS  15 

branches  of  mathematics  were  suggested  by  astronomical 
problems.  The  mathematical  processes  developed  for  astro- 
nomical applications  are,  of  course,  available  for  use  in 
other  fields.  But  the  great  science  of  mathematics  does 
not  exist  alone  for  its  applications,  and  to  have  stimulated 
its  growth  is  an  important  contribution.  While  many- 
parts  of  mathematics  did  not  have  their  origin  in  astro- 
nomical problems,  it  is  certain  that  had  it  not  been  for  these 
problems  mathematical  science  would  be  very  different  from 
what  it  now  is. 

The  science  of  dynamics  is  based  on  the  laws  of  motion. 
These  laws  were  first  completely  formulated  by  Newton, , 
who  discovered  them  and  proved  their  correctness  by  con- 
sidering the  revolutions  of  the  moon  and  planets,  which 
describe  their  orbits  under  the  ideal  condition  of  motion  in  a 
vacuum  without  any  friction.  The  immense  importance 
of  mechanics  in  modern  life  is  a  measure  of  the  value  of  this 
contribution  of  astronomy  to  science. 

The  science  of  geography  owes  much  to  astronomy,  both 
directly  and  indirectly.  A  great  period  of  exploration  fol- 
lowed the  voyages  of  Columbus.  It  took  courage  of  the 
highest  order  to  sail  for  many  weeks  over  an  unknown  ocean 
in  the  frail  boats  of  his  time.  He  had  good  reasons  for  think- 
ing he  could  reach  India,  to  the  eastward,  by  saDing  west- 
ward from  Spain.  His  reasons  were  of  an  astronomical 
nature.  He  had  seen  the  sun  rise  from  the  ocean  in  the 
east,  travel  across  the  sky  and  set  in  the  west ;  he  had  ob- 
served that  the  moon  and  stars  have  similar  motions ;  and 
he  inferjed  from  these  things  that  the  earth  was  of  finite  ex- 
tent and  that  the  heavenly  bodies  moved  around  it.  This 
led  him  to  believe  it  could  be  circumnavigated.  Eelying 
upon  the  conclusions  that  he  drew  from  his  observations  of 
the  motions  of  the  heavenly  bodies,  he  maintained  control 
of  his  mutinous  sailors  during  their  perilous  voyage  across 
the  Atlaptic,  and  made  a  discovery  that  has  been  of  immense 
consequence  to  the  human  race. 


16  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  i,  6 

One  of  the  most  important  influences  in  modern  scientific 
thought  is  the  doctrine  of  evolution.  It  has  not  only  largely- 
given  direction  to  investigations  and  speculations  in  biology 
and  geology,  but  it  has  also  been  an  important  factor  in  the 
interpretation  of  history,  social  changes,  and  even  religion. 
The  first  clear  ideas  of  the  orderly  development  of  the  uni- 
verse were  obtained  by  contemplating  the  relatively  simple 
celestial  phenomena,  and  the  doctrine  of  evolution  was  cur- 
rent in  astronomical  hterature  more  than  haK  a  century 
before  it  appeared  in  the  writings  of  Darwin,  Spencer,  and 
their  contemporaries.  In  fact,  it  was  carried  directly  from 
astronomy  over  into  geology,  and  from  geology  into  the 
biological  sciences  (Art.  242). 

7.  The  Present  Value  of  Astronomy.  —  From  what  has 
been  said  it  will  be  admitted  that  astronomy  has  been  of 
great  importance  in  the  development  of  science,  but  it  is 
commonly  believed  that  at  the  present  time  it  is  of  little 
practical  value  to  mankind.  While  its  uses  are  by  no 
means  so  numerous  as  those  of  physics  and  chemistry,  it 
is  nevertheless  quite  indispensable  in  a  number  of  human 
activities. 

Safe  navigation  of  the  seas  is  absolutely  dependent  upon 
astronomy.  In  all  long  voyages  the  captains  of  vessels 
frequently  determine  their  positions  by  observations  of  the 
celestial  bodies.  Sailors  use  the  nautical  mile,  or  knot, 
which  approximately  equals  one  and  one  sixth  ordinary 
miles.  The  reason  they  employ  the  nautical  mile  is  that  this 
is  the  distance  which  corresponds  to  a  change  of  one  minute 
of  arc  in  the  apparent  positions  of  the  heavenly  bodies. 
That  is,  if,  for  simplicity,  the  sun  were  over  a  meridian,  its 
altitude  as  observed  from  two  vessels  a  nautical  mile  apart 
on  that  meridian  would  differ  by  one  minute  of  arc. 

Navigation  is  not  only  dependent  on  simple  observations 
of  the  sun,  moon,  and  stars,  but  the  mathematical  theory 
of  the  motions  of  these  bodies  is  involved.  The  subject  is 
so  difficult  and  intricate  that  for  a  long  time  England  and 


CH.  I,  7]        PRELIMINARY  CONSIDERATIONS  17 

France  offered  substantial  cash  prizes  for  accurate  tables  of 
the  positions  of  the  moon  for  the  use  of  their  sailors. 

Just  as  a  sea  captain  deternaines  his  position  by  astro- 
nomical observations,  so  also  are  geographical  positions 
located.  For  example,  explorers  of  the  polar  regions  find 
how  near  they  have  approached  to  the  pole  by  observations 
of  the  altitude  of  the  sun.  International  boundary  lines  in 
many  cases  are  defined  by  latitudes  and  longitudes,  instead 
of  being  determined  by  natural  barriers,  as  rivers,  and  in  all 
such  cases  they  are  located  by  astronomical  observations. 

It  might  be  supposed  that  even  though  astronomy  is  essen- 
tial to  navigation  and  geography,  it  has  no  value  in  the 
ordinary  activities  of  life.  Here,  again,  first  impressions  are 
erroneous.  It  is  obvious  that  railway  trains  must  be  run  ac- 
cording to  accurate  time  schedules  in  order  to  avoid  confusion 
and  wrecks.  There  are  also  many  other  things  in  which  accurate 
time  is  important.  Now,  time  is  determined  by  observations 
of  the  stars.  The  miUions  of  clocks  and  watches  in  use  in 
the  world  are  all  ultimately  corrected  and  controlled  by 
comparing  them  with  the  apparent  diurnal  motions  of  the 
stars.  For  example,  in  the  United  States,  observations  are 
made  by  the  astronomers  of  the  Naval  Observatory,  at 
Washington,  on  every  clear  night,  and  from  these  observa- 
tions their  clocks  are  corrected.  These  clocks  are  in  elec-* 
trical  connection  with  more  than  30,000  other  clocks  in 
various  parts  of  the  country.  Every  day  time  signals  are 
sent  out  from  Washington  and  these  30,000  clocks  are 
automatically  corrected,  and  all  other  timepieces  are 
directly  or  indirectly  compared  with  them. 

It  might  be  inquired  whether  some  other  means  might 
not  be  devised  of  measuring  time  accurately.  It  might  be 
supposed  that  a  clock  coiild  be  made  that  would  run  so 
accurately  as  to  serve  all  practical  purposes.  The  fact  is, 
however,  no  clock  ever  wd,s  made  which  ran  accurately  for 
any  considerable  length  of  time.  No  two  clocks  have  been 
made  which  ran  exactly  alike.     In  order  to  obtain  a  satis- 

0 


18     AN  INTRODUCTION  TO  ASTRONOMY  [ch.  i,  7 

factory  measure  of  time  it  is  necessary  to  secure  the  ideal 
conditions  under  which  the  earth  rotates  and  the  heavenly 
bodies  move,  and  there  is  no  prospect  that  it  ever  will  be 
possible  to  use  anything  else,  as  the  fundamental  basis,  than 
the  apparent  motions  of  the  stars. 

Astronomy  is,  and  will  continue  to  be,  of  great  importance 
in  connection  with  other  sciences.  It  suppUes  most  of  the 
fundamental  facts  on  which  meteorology  depends.  It  is 
of  great  value  to  geology  because  it  furnishes  the  geologist 
information  respecting  the  origin  and  pre-geologic  history 
of  the  earth,  it  determines  for  him  the  size  and  shape  of  the 
earth,  it  measures  the  mass  of  the  earth,  and  it  proves  impor- 
tant facts  respecting  the  condition  of  the  earth's  interior. 
It  is  valuable  in  physics  and  chemistry  because  the  imiverse 
is  a  great  laboratory  which,  with  modem  instruments,  can 
be  brought  to  a  considerable  extent  within  reach  of  the 
investigator.  For  example,  the  sun  is  at  a  higher  tempera- 
ture than  can  be  produced  by  any  known  means  on  the 
earth.  The  material  of  which  it  is  composed  is  in  an  incan- 
descent state,  and  the  study  of  the  light  received  from  it  has 
proved  the  existence,  in  a  number  of  instances,  of  chemical 
elements  which  had  not  been  known  on  the  earth.  In  fact, 
their  discovery  in  the  sun  led  to  their  detection  on  the  earth. 
It  seems  probable  that  similar  discoveries  will  be  made  many 
times  in  the  future.  The  sun's  corona  and  the  nebulse 
contain  material  which  seems  to  be  in  a  more  primitive  state 
than  any  known  on  the  earth,  and  the  revelations  afforded 
by  these  objects  are  having  a  great  influence  on  physical 
theories  respecting  the  ultimate  structure  of  matter. 

Astronomy  is  of  greatest  value  to  mankind,  however,  in 
an  intellectual  way.  It  furnishes  men  with  an  idea  of  the 
wonderful  universe  in  which  they  live  and  of  their  position 
in  it.  Its  effects  on  them  are  analogous  to  those  which  are 
produced  by  travel  on  the  earth.  If  a  man  visits  various 
countries,  he  learns  many  things  which  he  does  not  and  can- 
not apply  on  his  return  home,   but  which,  nevertheless, 


CH.  I,  8]        PRELIMINARY  CONSIDERATIONS  19 

make  him  a  broader  and  better  man.  Similarly,  though 
what  one  may  learn  about  the  millions  of  worlds  which 
occupy  the  almost  infinite  space  within  reach  of  the  great 
telescopes  of  modern  times  cannot  be  directly  applied  in  the 
ordinary  affairs  of  life,  yet  the  contemplation  of  such  things, 
in  which  there  is  never  anything  that  is  low  or  mean  or  sordid, 
makes  on  him  a  profound  impression.  It  strongly  modifies 
the  particular  philosophy  which  he  has  more  or  less  definitely 
formulated  in  his  consciousness,  and  in  harmony  with  which 
he  orders  his  Ufe. 

8.  The  Scope  of  Astronomy.  —  The  popular  conception 
of  astronomy  is  that  it  deals  in  some  vague  and  speculative 
way  with  the  stars.  Since  it  is  obviously  impossible  to 
visit  them,  it  is  supposed  that  all  conclusions  respecting  them, 
except  the  few  facts  revealed  directly  by  telescopes,  are  pure 
guesses.  Many  people  suppose  that  astronomers  ordinarily 
engage  in  the  harmless  and  useless  pastime  of  gazing  at  the 
stars  with  the  hope  of  discovering  a  new  one.  Many  of  those 
who  do  not  have  this  view  suppose  that  astronomers  control 
the  weather,  can  tell  fortunes,  and  are  very  shrewd  to  have 
discovered  the  names  of  so  many  stars.  As  is  true  of  most 
conclusions  that  are  not  based  on  evidence,  these  conceptions 
of  astronomy  and  astronomers  are  absurd. 

Astronomy  contains  a  great  mass  of  firmly  established 
facts.  Astronomers  demand  as  much  evidence  in  support 
of  their  theories  as  is  required  by  other  scientists.  They 
have  actually  measured  the  distances  to  the  moon,  sun,  and 
many  of  the  stars.  They  have  discovered  the  laws  of  their 
motions  and  have  determined  the  masses  of  the  principal 
members  of  the  solar  system.  The  precision  attained  in 
much  of  their  work  is  beyond  that  realized  in  most  other 
sciences,  and  their  greatest  interest  is  in  measurable  things 
and  not  in  vague  speculations. 

A  more  extended  preliminary  statement  of  the  scope  of 
astronomy  is  necessary  in  order  that  its  study  may  be  entered 
on  without  misunderstandings.     Besides,  the  relations  among 


20 


AN   INTRODUCTION    TO   ASTRONOMY     [ch.  i,  8 


the  facts  with  which  a  science  deals  are  very  important, 
and  a  preliminary  outUne  of  the  subject  will  make  it  easier 
to  place  in  their  proper  position  in  an  organized  whole  all 
the  various  things  which  may  be  set  forth  in  the  detailed 
discussions. 

The  most  accessible  and  best-known  astronomical  object 
is  the  earth.     Those  facts  respecting  it  that  are  determined 

entirely  or  in  large 
part  by  astronomical 
means  are  properly 
regarded  as  belong- 
ing to  astronomy. 
Among  them  are  the 
shape  and  size  of 
the  earth,  its  average 
density,  the  condition 
of  its  interior,  the 
height  of  its  atmos- 
phere, its  rotation  on 
its  axis  and  revolu- 
tion around  the  sun, 
and  the  climatic  con- 
ditions of  its  surface 
so  far  as  they  are 
determined  by  its  re- 
lation to  the  sun. 

The  nearest  celes- 
tial body  is  the 
moon.  Astronomers 
have  found  by  fundamentally  the  same  methods  as  those 
which  surveyors  employ  that  its  distance  from  the  earth 
averages  about  240,000  miles,  that  its  diameter  is  about 
2160  miles,  and  that  its  mass  is  about  one  eightieth  that  of 
the  earth.  The  earth  holds  the  moon  in  its  orbit  by  its  gravi- 
tational control,  and  the  moon  in  turn  causes  the  tides  on  the 
earth.     It  is  found  that  there  is  neither  atmosphere  nor  water 


Fig.  4.  —  The  moon  1.5  days  after  the  first 
quarter.  Photographed  with  the  40~inch 
telescope  of  the   Yerkes  Observatory. 


CH.  I,  8]         PEELIMINARY   CONSIDERATIONS  21 

on  the  moon,  and  the  telescope  shows  that  its  surface  is 
covered  with  mountains  and  circular  depressions,  many  of 
great  size,  which  are  called  craters. 

The  earth  is  one  of  the  eight  planets  which  revolve  around 
the  sun  in  nearly  circular  orbits.  '  Three  of  them  are  smaller 
than  the  earth  and  four  are  larger.  The  smallest,  Mercury, 
has  a  volume  about  one  twentieth  that  of  the  earth,  and  the 
largest,  Jupiter,  has  a  volume  about  one  thousand  times  that 
of  the  earth.  The  great  sim,  whose  mass  is  seven  himdred 
times  that  of  all  of  the  planets  combined,  holds  them  in  their 
orbits  and  lights  and  warms  them  with  its  abundant  rays. 
Those  nearest  the  sun  are  heated  much  more  than  the  earth, 
but  remote  Neptune  gets  only  one  nine-hundredth  as  much 
light  and  heat  per  unit  area  as  is  received  by  the  earth. 
Some  of  the  planets  have  no  moons  and  others  have  several. 
The  conditions  on  one  or  two  of  them  seem  to  be  perhaps 
favorable  for  the  development  of  life,  while  the  others  cer- 
tainly cannot  be  the  abode  of  such  life  as  flourishes  on  the 
earth. 

In  addition  to  the  planets,  over  eight  hundred  small 
planets,  or  planetoids,  and  a  great  number  of  comets  circu- 
late around  the  sun  in  obedience  to  the  same  law  of  gravita- 
tion. The  orbits  of  nearly  all  the  small  planets  lie  between 
the  orbits  of  Mars  and  Jupiter ;  the  orbits  of  the  comets  are 
generally  very  elongated  and  are  unrelated  to  the  other 
members  of  the  system.  The  phenomena  presented  by  the 
comets,  for  example  the  behavior  of  their  tails,  raise  many 
interesting  and  puzzling  questions. 

The  dominant  member  of  the  solar  system  is  the  sun. 
Its  volume  is  more  than  a  million  times  that  of  the  earth, 
its  temperature  is  far  higher  than  any  that  can  be  produced 
on  the  earth,  even  in  the  most  efficient  electrical  furnaces, 
and  its  surface  is  disturbed  by  the  most  violent  storms. 
Often  masses  of  this  highly  heated  material,  in  volumes 
greater  than  the  whole  earth,  move  along  or  spout  up  from 
its  surface  at  the  rate  of  several  hundreds  miles  a  minute. 


22      AN  INTRODUCTION  TO  ASTRONOMY  [ch.  i,  8 

The  spectroscope  shows  that  the  sun  contains  many  of  the 
elements,  particularly  the  metals,  of  which  the  earth  is  com- 
posed. The  consideration  of  the  possible  sources  of  the 
sun's  heat  leads  to  the  conclusion  that  it  has  supplied  the 
earth  with  radiant  energy  for  many  milUons  of  years,  and 
that  the  supply  will  not  fail  for  at  least  a  number  of  milUon 
years  in  the  future. 

The  stars  that  seem  to  fill  the. sky  on  a  clear  night  are 
Sims,  many  of  which  are  much  larger  and  more  brilliant  than 
our  own  sun.  They  appear  to  be  relatively  faint  points  of 
light  because  of  their  enormous  distances  from  us.  The 
nearest  of  them  is  so  remote  that  more  than  four  years  are 
required  for  its  hght  to  come  to  the  solar  system,  though 
Ught  travels  at  the  rate  of  186,330  miles  per  second;  and 
others,  still  within  the  range  of  large  telescopes,  are  certainly 
a  thousand  times  more  distant.  At  these  vast  distances 
such  a  tiny  object  as  the  earth  would  be  entirely  invisible 
even  though  astronomers  possessed  telescopes  ten  thousand 
times  as  powerful  as  those  now  in  use.  Sometimes  stars 
appear  to  be  close  together,  as  in  the  case  of  the  Pleiades,  but 
their  apparent  proximity  is  due  to  their  immense  distances 
from  the  observer.  There  are  doubtless  regions  of  space 
from  which  the  sun  would  seem  to  be  a  small  star  forming  a 
close  group  with  a  number  of  others.  There  are  visible 
to  the  unaided  eye  in  all  the  sky  only  about  5000  stars,  but 
the  great  photographic  telescopes  with  which  modem 
observatories  are  equipped  show  several  hundreds  of  millions 
of  them.  It  might  be  supposed  that  telescopes  with  twice 
the  light-gathering  power  would  show  proportionately  more 
stars,  and  so  on  indefinitely,  but  this  is  certainly  not  true, 
for  there  is  evidence  that  points  to  the  conclusion  that  they 
do  not  extend  indefinitely,  at  least  with  the  frequency  with 
which  they  occur  in  the  region  around  the  sun.  The  visible 
stars  are  not  uniforinly  scattered  throughout  the  space  which 
they  occupy,  but  form  a  great  disk-like  aggregation  lying  in 
the  plane  of  the  Milky  Way. 


CH.  I,  8]         PRELIMINARY   CONSIDERATIONS  23 

Many  stars,  instead  of  being  single  isolated  masses,  like 
the  sun,  are  found  on  examination  with  highly  magnifjdng 
telescopes  to  consist  of  two  suns  revolving  around  their 
common  center  of  gravity.  In  most  cases  the  distances 
between  the  two  members  of  a  double  star  is  several  times 
as  great  as  the  distance  from  the  earth  to  the  sun.  The 
existence  of  double  stars  which  may  be  much  closer  together 
than  those  which  are  visible  through  telescopes  has  also 
been  shown  by  means  of  instruments  called  spectroscopes. 
It  has  been  found  that  a  considerable  fraction,  probably 
one  fourth,  of  all  the  nearer  stars  are  double  stars.  There 
are  also  triple  and  quadruple  stars;  and  in  some  cases 
thousands  of  suns,  all  invisible  to  the  unaided  eye,  occupy 
a  part  of  the  sky  apparently  smaller  than  the  moon.  Even 
in  such  cases  the  distances  between  the  stars  are  enormous, 
and  such  clusters,  as  they  are  called,  constitute  larger  and 
more  wonderful  aggregations  of  matter  than  any  one  ever 
dreamed  existed  before  they  were  revealed  by  modern 
instruments. 

While  the  sun  is  the  center  around  which  the  planets  and 
comets  revolve,  it  is  not  fixed  with  respect  to  the  other 
stars.  Observations  with  both  the  telescope  and  the  spec- 
troscope prove  that  it  is  moving,  with  respect  to  the  brighter 
stars,  approximately  in  the  direction  of  the  brilliant  Vega 
in  the  constellation  Lyra.  It  is  found  by  use  of  the  spectro- 
scope that  the  rate  of  motion  is  about  400,000,000  miles 
per  year.  The  other  stars  are  also  in  motion  with  an  average 
velocity  of  about  600,000,000  miles  per  year,  though  some  of 
them  move  much  more  slowly  than  this  and  some  of  them 
many  times  faster.  One  might  think  that  the  great  speed  of 
the  sun  would  in  a  century  or  two  so  change  its  relations  to 
the  stars  that  the  appearance  of  the  sky  would  be  entirely 
altered.  But  the  stars  are  so  remote  that  in  comparison  the 
distance  traveled  by  the  sun  in  a  year  is  neghgible.  When 
those  who  built  the  pyramids  turned  their  eyes  to  the  sky 
at  night  they  saw  the  stars  grouped  in  constellations  almost 


24  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  i,  8 

exactly  as  they  are  seen  at  present.  During  the  time  cov- 
ered by  observations  accurate  enough  to  show  the  motion 
of  the  sun  it  has  moved  sensibly  in  a  straight  line,  though  in 
the  course  of  time  the  direction  of  its  path  will  doubtless  be 
changed  by  the  attractions  of  the  other  stars.  Similarly, 
the  other  stars  are  moving  in  sensibly  straight  lines  in  every 
direction,  but  not  altogether  at  random,  for  it  has  been  found 
that  there  is  a  general  tendency  for  them  to  move  in  two  or 
more  roughly  parallel  streams. 

In  addition  to  learning  what  the  imiverse  is  at  present, 
one  of  the  most  important  and  interesting  objects  of  astron- 
omy is  to  find  out  through  what  great  series  of  changes  it 
has  gone  in  its  past  evolution,  and  what  will  take  place  in  it 
in  the  future.  As  a  special  problem,  the  astronomer  tries 
to  discover  how  the  earth  originated,  how  long  it  has  been 
in  existence,  particularly  in  a  state  adapted  to  the  abode  of 
life,  and  what  reasonably  may  be  expected  for  the  future. 
These  great  problems  of  cosmogony  have  been  of  deep  inter- 
est to  mankind  from  the  dawn  of  civilization ;  with  increasing 
knowledge  of  the  wonders  of  the  universe  and  of  the  laws 
by  which  alone  such  questions  can  be  answered,  they  have 
become  more  and  more  absorbingly  attractive. 

I.  QUESTIONS 

1.  Enumerate  as  many  ways  as  possible  in  which  science  is 
beneficial  to  men. 

2.  What  is  the  fmidamental  basis  on  which  science  rests,  and 
what  are  its  chief  characteristics  ? 

3.  What  is  induction  ?  Give  examples.  Can  a  science  be  de- 
veloped without  inductions  ?    Are  inductions  always  true  ? 

4.  What  is  deduction?  Give  examples.  Can  a  science  be  de- 
veloped without  deductions  ?    Are  deductions  always  true  ? 

5.  In  what  respects  may  science  be  imperfect  ?  How  may  its  im- 
perfections be  most  largely  eUminated  ?  Are  any  human  activities 
perfect  ? 

6.  Name  some  superstition  and  show  in  what  respects  It  differs 
from  scientific  conclusions. 

7.  Why  did  science  originate  in  astronomy  ? 


CH.  I,  8]         PRELIMINARY   CONSIDERATIONS  25 

8.  Are  conclusions  in  astronomy  firmly  established,  as  they  are 
in  other  sciences  ?  .  ■ 

9.  In  what  fundamental  respects  do  scientific  laws  differ  from 
civil  laws  ? 

10.  What  advantages  may  be  derived  from  a  preliminary  outUne 
of  the  scope  of  astronomy  ?  Would  they  hold  in  the  case  of  a  sub- 
ject not  a  science  ? 

11.  What  questions  respecting  the  earth  are  properly  regarded 
as  belonging  to  astronomy?  To  what  other 'sciences  do  they  re- 
spectively belong  ?  Is  there  any  science  which  has  no  common 
ground  with  some  other  science  ? 

12.  What  arts  are  used  in  astronomy?  Does  astronomy  con- 
tribute to  any  art  ? 

13.  What  references  to  astronomy  in  the  sacred  or  classical  htera- 
tures  do  you  know  ? 

14.  Has  astronomy  exerted  any  influence  on  philosophy  and 
rehgion  ?    Have  they  modified  astronomy  ? 


CHAPTER  II 
THE   EARTH 

I.   The  Shape  of  the  Eakth 

9.  Astronomical  Problems  respecting  the  Earth.  —  The 

earth  is  one  of  the  objects  belonging  to  the  field  of  astronom- 
ical investigations.  In  the  consideration  of  it  astronomy 
has  its  closest  contact  with  some  pf  the  other  sciences,  par- 
ticularly with  geology  and  meteorology.  Those  problems 
respecting  the  earth  that  can  be  solved  for  other  planets  also, 
or  that  are  essential  for  the  investigation  of  other  astronom- 
ical questions,  are  properly  considered  as  belonging  to  the 
field  of  astronomy. 

The  astronomical  problems  respecting  the  earth  can  be 
divided  into  two  general  classes.  The  first  class  consists  of 
those  which  can  be  treated,  at  least  to  a  large  extent,  with- 
out regarding  the  earth  as  a  member  of  a  family  of  planets 
or  considering  its  relations  to  them  and  the  sun.  Such  prob- 
lems are  its  shape  and  size,  its  mass,  its  density,  its  interior 
temperature  and  rigidity,  and  the  constitution,  mass,  height, 
and  effects  of  its  atmosphere.  These  problems  will  be  treated 
in  this  chapter.  The  second  class  consists  of  the  problems 
involved  in  the  relations  of  the  earth  to  other  bodies,  partic- 
ularly its  rotation,  revolution  around  the  sun,  and  the  con- 
sequences of  these  motions.  The  treatment  of  these  prob- 
lems will  be  reserved  for  the  next  chapter. 

It  would  be  an  easy  matter  simply  to  state  the  astronom- 
ical facts  respecting  the  earth,  but  in  science  it  is  necessary 
not  only  to  say  what  things  are  true  but  also  to  give  the 
reasons  for  believing  that  they  are  true.  Therefore  one  or 
more  proofs  will  be  given  for  the  conclusions  astronomers 
have  reached  respecting  the  earth.     As  a  matter  of  logic 

26 


CH.  II,  10] 


THE   EARTH 


27 


one  complete  proof  is  sufficient,  but  it  must  be  remembered 
that  a  scientific  doctrine  consists  of,  and  rests  on,  a  great 
number  of  theories  whose  truth  may  be  more  or  less  in  ques- 
tion, and  consequently  a  number  of  proofs  is  always  desir- 
able. If  they  agree,  their  agreement  confirms  belief  in  the 
accuracy  of  all  of  them.  It  will  not  be  regarded  as  a  burden 
to  follow  carefully  these  proofs ;  in  fact,  one  who  has  arrived 
at  a  mature  stage  of  intellectual  development  instinctively 
demands  the  reasons  we  have  for  believing  that  our  conclu- 
sions are  sound. 

10.  The   Simplest  and  most  Conclusive  Proof  of    the 
Earth's  Sphericity.^  —  Among  the  proofs  that  the  earth  is 

round,  the  simplest  and  most  concliisive  is  that  the  plane  of 

the  horizon,  or  the  direction  of  the  plumb  line,  changes  by  an 

angle  which  is  direcRy  proportional 

to-tlhe  distance  the  observer  travels 

along   the   surface   of   the   earth, 

whatever  the  direction  and  distance 

of  travel. 

It  will  be  shown  first  that  if 

the  earth  were  a  true  sphere  the 

statement  would  be  true.      For 

simphcity,  suppose  the  observer 

travels  along  a  meridian.    If  the 

statement  is  true  for  this  case, 

it  will    be  true   for   all   others, 

because  a  sphere  has  the  same 

curvature    in    every    direction. 

Suppose  the  observer  starts  from 

Oi,  Fig.  5,  and  travels  northward 

to  O2.     The  length   of  the   arc 

O1O2  is  proportional  to  the  angle 

a  which  it  subtends  at  the  center  of  the  sphere.    The  planes 

of  the  horizon  of  Oi  and  O2  are  respectively  OJIi  and  OoBi. 

'  The  earth-  is  not  exactly^ound,  but  the  departure  from  sphericity 
will  be  neglected  for  the  moment. 


Fig.  5.  —  The  change  in  the  di- 
rection ■  of  the  plumb  line  is 
proportional  to  the  distance 
traveled  along  the  surface  of 
the  earth. 


28  AN  INTRODUCTION  TO  ASTRONOMY     [ch.  ii,  10 

These  lines  are  respectively  perpendicular  to  COi  and  CO2. 
Therefore  the  angle  between  them  equals  the  angle  a.  That 
is,  the  distance  traveled  is  proportional  to  the  change  of 
direction  of  the  plane  of  the  horizon. 

The  plumb  lines  at  Oi  and  O2  are  respectively  OiZi  and 
O2Z2,  and  the  angle  between  these  hues  is  a.  Hence  the  dis- 
tance traveled  is  proportional  to  the  change  in  the  direction 
of  the  plumb  Une. 

It  will  be  shown  now  that  if  the  surface  of  the  earth  were 
not  a  true  sphere  the  change  in  the  direction  of  the  plane  of 
the  horizon  would  not  be  proportional  to  the  distance  traveled 

on  the  surface.  Suppose 
Fig.  6  represents  a  plane 
section  through  the  non- 
spherical  earth  along 
whose  surface  the  ob- 
server travels.  Since  the 
earth  is  not  a  sphere,  the 
curvature  of  its  surface 
will  be  different  at  differ- 
ent places.    Suppose  that 

Fig.  6. -If  the  earth  were  not  spherical,  ^lO^  IS  One  of  the  flatter 
equal  angles  would  be  subtended  by  arcs  regions  and  O3O4  is  one 
of  different  lengths.  r  .1 

01  the  more  convex  ones. 
In  the  neighborhood  of  O1O2  the  direction  of  the  plumb  Une 
changes  slowly,  while  in  the  neighborhood  of  O3O4  its  direc- 
tion changes  more  rapidly.  The  large  arc  O1O2  subtends  an 
angle  at  Ci  made  by  the  respective  perpendiculars  to  the 
surface  which  exactly  equals  the  angle  at  C3  subtended  by 
the  smaller  arc  O3O4.  Therefore  in  this  case  the  change  in 
direction  of  the  plumb  line  is  not  proportional  to  the  dis- 
tance traveled,  for  the  same  angular  change  corresponds  to 
two  different  distances.  The  same  result  is  true  for  the 
plahe  of  the  horizon  because  it  is  always  perpendicular  to 
the  plumb  line. 

Since  the  conditions  of  the  statement  would  be  satisfied 


CH.  n,  lO] 


THE   EARTH 


29 


in  case  the  earth  were  spherical,  and  only  in  case  it  were 
spherical,  the  next  question  is  what  the  observations  show. 
Except  for  irregularities  of  the  surface,  which  are  not  under 
consideration  here,  and  the  oblateness,  which  will  be  dis- 
cussed in  Art.  12,  the  observations  prove  absolutely  that  the 
change  in  direction  of  the  plumb  line  is  proportional  to  the 
arc  traversed. 

Two  practical  problems  are  involved  in  carrying  out  the 
proof  which  has  just  been  described.  The  first  is  that  of 
measuring  the  distance  between  two  points  along  the  sur- 


FiG.  7. - 


-  The  base  line  A\Ai  is  measured  directly  and  the  other  distances 
are  obtained  by  triangulation. 


face  of  the  earth,  and  the  second  is  that  of  determining  the 
change  in  the  direction  of  the  plumb  line.  The  first  is  a 
refined  problem  of  surveying;  the  second  is  solved  by 
observations  of  the  stars. 

All  long  distances  on  the  surface  of  the  earth  are  deter- 
mined by  a  process  known  as  triangulation.  It  is  much 
more  convenient  than  direct  measurement  and  also  much 
more  accurate.  A  fairly  level  stretch  of  country,  Ai  and 
A2  in  Fig.  7,  a  few  miles  long  is  selected,  and  the  distance 
between  the  two  points,  which  must  be  visible  from  each 
other,   is   measured   with   the  greatest   possible   accuracy. 


30  AN  INTRODUCTION   TO  ASTRONOMY     [ch.  ii,  10 

This  line  is  called  the  base  line.  Then  a  point  A3  is  taken 
which  can  be  seen  from  both  Ai  and  A^.  A  telescope  is  set 
up  at  Ai  and  pointed  at  A2.  It  has  a  circle  parallel  to  the 
surface  of  the  earth  on  which  the  degrees  are;.marked.  The 
position  of  the  telescope  with  respect  to  this  circle  is  recorded. 
Then  the  telescope  is  turned  until  it  points  toward  A3. 
The  difference  of  its  position  with  respect  to  the  circle  when 
pointed  at  A2  and  at  A3  is  the  angle  A2A1A3.  Similarly, 
the  telescope  is  set  up  at  A2  and  the  angle  A1A2A3  is  meas- 
ured. Then  in  the  triangle  A1A2A3  two  angles  and  the  in- 
cluded side  are  known.  By  plane  geometry,  two  triangles 
that  have  two  angles  and  the  included  side  of  one  respectively 
equal  to  two  angles  and  the  included  side  of  the  other  are 
exactly  alike  in  size  and  shape.  This  simply  means  that 
when  two  angles  and  the  included  side  of  the  triangle  are 
given,  the  triangle  is  uniquely  defined.  The  remaining  parts 
can  be  computed  by  trigonometry.  In  the  present  case 
suppose  the  distance  A2A3  is  computed. 

Now  suppose  a  fourth  point  A  4  is  taken  so  that  it  is 
visible  from  both  A2  and  A3.  Then,  after  the  angles  at  A2 
and  A3  in  the  triangle  A2A3A4  have  been  measured,  the  line 
A3A4  can  be  computed.  This  process  evidently  can  be  con- 
tinued, step  by  step,  to  any  desired  distance. 

Suppose  Ai  is  regarded  as  the  original  point  from  which 
measurements  are  to  be  made.  Not  only  have  various  dis- 
tances been,  determined,  but  also  their  directions  with  respect 
to  the  north-south  line  are  known.  Consequently,  it  is 
known  how  far  north  and  how  far  east  A 2  is  from  Ai.  The 
next  step  gives  how  far  south  and  how  far  east  A3  is  from  A2. 
By  combining  the  two  results  it  is  known  how  far  south  and 
how  far  east  A3  is  from  Ai,  and  so  on  for  succeeding  points. 

The  convenience  in  triangulation  results  partly  from  the 
long  distances  that  can  be  measured,  especially  in  rough 
country.  It  is  sometimes  advisable  to  go  to  the  trouble  of 
erecting  towers  in  order  to  make  it  possible  to  use  stations 
separated  by  long  distances.     The  accuracy  arises,  at  least 


CH.  11,  12]  THE   EARTH  31 

in  part,  from  the  fact  that  the  angles  are  measured  by  in- 
struments which  magnify  them.  The  fact  that  the  stations 
are  not  all  on  the  same  level,  and  the  curvature  of  the  earth, 
introduce  little  difBculties  in  the  computations  that  must 
be  carefully  overcome. 

The  direction  of  the  plumb  Una  at  the  station  Ai,  for 
example,  is  determined  by  noting  the  point  among  the  stars 
at  which  it  points.  The  plumb  line  at  A2  will  point  to  a 
dififerent  place  among  the  stars.  The  difference  in  the  two 
places  among  the  stars  gives  the  difference  in  the  directions 
of  the  plumb  lines  at  the  two  stations.  The  stars  apparently 
move  across  the  sky  from' east  to  west  during  the  night  and 
are  not  in  the  same  positions  at  the  same  time  of  the  day 
on  different  nights.  Hence,  there  are  here  also  certain  cir- 
cumstances to  which  careful  attention  must  be  given  in 
order  to  get  accurate  results. 

11.  Other  Proofs  of  the  Earth's  Sphericity.  —  There  are 
many  reasons  given  for  believing  that  the  earth  is  not  a 
plane,  and  that  it  is,  indeed,  some  sort  of  a  convex  figure ; 
but  most  of  them  do  not  prove  that  it  is  actually  spherical. 
It  will  be  sufficient  to  mention  them. 

(a)  The  earth  has  been  circumnavigated,  but  so  far  as 
this  fact  alone  is  concerned  it  might  be  the  shape  of  a  cu- 
cumber. (6)  Vessels  disappear  below  the  horizon  hulls  first 
and  masts  last,  but  this  only  proves  the  convexity  of  the 
surface,  (c)  The  horizon  appears  to  be  a  circle  when  viewed 
from  an  elevation  above  the  surface  of  the  water,  This  is 
theoretically  good  but  observationally  it  is  not  very  exact. 
(d)  The  shadow  of  the  earth  on  the  moon  at  the  time  of  a 
lunar  eclipse  is  always  an  arc  of  a  circle,  but  this  proof  is 
very  inconclusive,  in  spite  of  the  fact  that  it  is  often  men- 
tioned, because  the  shadow  has  no  very  definite  edge  and 
its  radius  is  large  compared  to  that  of  the  moon. 

12.  Proof  of  the  Oblateness  of  the  Earth  by  Arcs  of 
Latitude.  —  The  latitude  of  a  place  on  the  earth  is  deter- 
mined by  observations  of  the  direction  of  the  plumb  line 


32  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  ii,  12 

with  respect  to  the  stars.  This  is  the  reason  that  a  sea  cap- 
tain refers  to  the  heavenly  bodies  in  order  to  find  his  loca- 
tion on  the  ocean.  It  is  found  by  actual  observations  of  the 
stars  and  measurements  of  arcs  that  the  length  of  a  degree 
of  arc  is  longer  the  farther  it  is  from  the  earth's  equator. 
This  proves  that  the  earth  is  less  curved  at  the  poles  than 
it  is  at  the  equator.  A  body  which  is  thus  flattened  at  the 
poles  and  bulged, at  the  equator  is  called  oblate. 

In  order  to  see  that  in  the  case  of  an  oblate  body  a  degree 
of  latitude  is  longer  near  the  poles  than  it  is  at  the  equator, 
consider  Fig.  8.     In  this  figure  E  represents  a  plane  section 

of  the  body  through  its  poles. 
The  curvature  at  the  equator  is 
the  same  as  the  curvature  of  the 
circle  Ci,  and  a  degree  of  latitiide 
<  on  S  at  its  equator  equals  a 
degree  of  latitude  on  Ci.  The 
curvature  of  E  at  its  pole  is  the 
same  as  the  curvature  of  the 
circle  d,  and  a  degree  of  lati- 

""-- .--'  tude  on  E  at  its  pole  equals  a 

I^G.  8.  —  The  length  of  a  degree    degree  of  latitude  on  C2.     Since 

t^:^d"ii^t^:ttt1he';or"    C^  i«  greater  than  C,  a  degree 

of  latitude  near  the  pole  of  the 
oblate  body  is  greater  than  a  degree  of  latitude  near  its 
equator. 

A  false  argument  is  sometimes  made  which  leads  to  the 
opposite  conclusion.  Lines  are  drawn  from  the  center  of 
the  oblate  body  dividing  the  quadrant  into  a  number  of 
equal  angles.  Then  it  is  observed  that  the  arc  intercepted 
between  the  two  fines  nearest  the  equator  is  longer  than 
that  intercepted  between  the  two  lines  nearest  the  pole. 
The  error  of  this  argument  lies  in  the  fact  that,  with  the 
exception  of  those  drawn  to  the  equator  and  poles,  these 
fines  are  not  perpendicular  to  the  surface.  Figure  9  shows 
an  oblate  body  with  a  number  of  lines  drajwn  perpendicular 


CH.  11,  13]  THE   EARTH  33 

to  its  surface.  Instead  of  their  all  passing  through  the 
center  of  the  body,  they  are  tangent  to  the  curve  AB.  The 
line  AE  equals  the  radius 
of  a  circle  having  the 
same  curvature  as  the 
oblate  body  at  E,  and 
BP  is  the  radius  of  the 
circle  having  the  curva- 
ture at  P. 

13.  Size  and  Shape 
of  the  Earth.  —  The  size 
and  shape  of  the  earth 
can  both  be  determined  „      „      „         ,.    , 

.           r  *i°-  »•  —  Perpendiculars  to  the  surface  of 

trom     measurements     ot  an  oblate  body,  showing  that  equal  arcs 

arcs.      If  the   earth  were  subtend  largest  angles  at  its  equator  and 

'  ,  smallest  at  its  poles. 

sphencal,  a  degree  of  arc 

would  have  the  same,  length  everywhere  on  its  surface,  and 
its  circumference  would  be  360  times  the  length  of  one  de- 
gree. Since  the  earth  is  oblate,  the  matter  is  not  quite  so 
simple.  But  from  the  lengths  of  arcs  in  different  latitudes 
both  the  size  and  the  shape  of  the  earth  can  be  computed. 

It  is  sufficiently  accurate  for  ordinary  purposes  to  state 
that  the  diameter  of  the  earth  is  about  8000  miles,  and  that 
the  difference  between  the  equatorial  and  polar  diameters  is 
27  miles. 

The  dimensions  of  the  earth  have  been  computed  with 
great  accuracy  by  Hayford,  who  found  for  the  equatorial 
diameter  7926.57  miles,  and  for  the  polar  diameter  7899.98 
miles.  The  error  in  these  results  cannot  exceed  a  thousand 
feet.  The  equatorial  circumference  is  24,901 .7  miles,  and  the 
length  of  one  degree  of  longitude  at  the  equator  is  69.17 
miles.  The  lengths  of  degrees  of  latitude  at  the  equator 
and  at  the  poles  are  respectively  69.40  and  68.71  miles. 
The  total  area  of  the  earth  is  about  196,400,000  square  miles. 
The  volume  of  the  earth  is  equal  to  the  volume  of  a  sphere 
whose  radius  is  3958.9  miles. 

D 


34  AN    INTRODUCTION   TO  ASTRONOMY     [ch.  ii,  14 


14.   Newton's  Proof  of  the  Oblateness  of  the  Earth.  — 

The  first  proof  that  the  earth  is  oblate  was  due  to  Newton. 
He  based  his  demonstration  on  the  laws  of  motion,  \the  law 
of  gravitation,  and  the  rotation-of  the  earth.  It  is  therefore 
much  more  compUcated  than  that  depending  on  the  lengths 
of  degrees  of  latitude,  which  is  purely  geometrical.  It  has 
the  advantage,  however,  of  not  requiring  any  measurements 
of  arcs. 

Suppose  the  earth,  Fig.  10,  rotates  around  the  axis  PP'. 
Imagine  that  a  tube  filled  with  water  exists  reaching  from 

the  pole  P  to  the  center 
C,  and  then  to  the  sur- 
face on  the  equator  at  Q. 
The  water  in  this  tube 
exerts  a  pressure  toward 
the  center  because  of  the 
attraction  of  the  earth 
for  it.  Consider  a  unit 
voliune  in  the  part  CP 
at  any  distance  D  from 
the  center ;  the  pressure 
it    exerts    toward     the 

Fig.   10.  —  Because  of  the  earth's  rotation    Center  equals  the  earth's 

around  PP'  the  column  CQ  must  be  attraction  for  it  because 

longer  than  PC.  . 

it  is  subject  to  no  other 
forces.  Suppose  for  the  moment  that  the  earth  is  a  sphere, 
as  it  would  be  if  it  were  not  rotating  on  its  axis,  and  con- 
sider a  unit  volume  in  the  part  CQ  at  the  distance  D  from 
the  center.  Because  of  the  symmetry  of  the  sphere  it 
will  be  subject  to  an  attraction  equal  to  that  on  the  corre- 
sponding unit  in  CP.  But,  in  addition  to  the  earth's  at- 
traction, this  mass  of  water  is  subject  to  the  centrifugal  force 
due  to  the  earth's  rotation,  which  to  some  extent  counter- 
balances the  attraction.  Therefore,  the  pressure  it  exerts 
toward  the  center  is  less  than  that  exerted  by  the  corre- 
sponding unit  in  CP-     If  the  earth  were  spherical,  all  units 


CH.  II,  15]  THE    EARTH  35 

in  the  two  columns  could  be  paired  in  this  way.  The  result 
would  be  that  the  pressure  exerted  by  PC  would  be  greater 
than  that  exerted  by  QC ;  but  such  a  condition  would  not 
be  one  of  equihbrium,  and  water  would  flow  out  of  the 
mouth  of  the  tube  from  the  center  to  the  equator.  In 
order  that  the  two  columns  of  water  shall  be  in  equihbrium 
the  equatorial  column  must  be  longer  than  the  polar. 

Newton  computed  the  amount  RQthy  which  the  one  tube 
must  be  longer  than  the  other  in  order  that  for  a  body  hav- 
ing the  mass,  dimensions,  and  rate  of  rotation  of  the  earth, 
there  should  be  equihbrium.  This  gave  him  the  oblate- 
ness  of  the  earth.  In  spite  of  the  fact  that  his  data  were 
not  very  exact,  he  obtained  results  which  agree  very  well 
with  those  furnished  by  modern  measurements  of  arcs. 

The  objection  at  once  arises  that  the  tubes  did  not 
actually  exist  and  that  they  could  not  possibly  be  constructed, 
and  therefore  that  the  conclusion  was  as  insecure  as  those 
usually  are  which  rest  on  imaginary  conditions.  But  the 
fears  aroused  by  these  objections  are  dissipated  by  a  little 
more  consideration  of  the  subject.  It  is  not  necessary  that 
the  tubes  should  run  in  straight  lines  from  the  surface  to 
the  center  in  order  that  the  principle  should  apply.  They 
might  bend  in  any  manner  and  the  results  would  be  the  same, 
just  as  the  level  to  which  the  water  rises  in  the  spout  of  a 
teakettle  does  not  depend  on  its  shape.  Suppose  the  tubes 
are  deformed  into  a  single  one  connecting  P  and  Q  along 
the  surface  of  the  earth.  The  principles  still  hold ;  but  the 
ocean  connection  of  pole  and  equator  may  be  considered  as 
being  a  tube.  Hence  the  earth  must  be  oblate  or  the  ocean 
would  flow  from  the  poles  toward  the  equator. 

15.  Pendulum  Proof  of  the  Oblateness  of  the  Earth.  — 
It  seems  strange  at  flrst  that  the  shape  of  the  earth  can  be 
determined  by  means  of  the  pendulum.  Evidently  the 
method  cannot  rest  on  such  simple  geometrical  principles  as 
were  sufl&cient  in  using  the  lengths  of  arcs.  It  will  be  found 
that  it  involves  the  laws  of  motion  and  the  law  of  gravitation. 


36  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  15 

The  time  of  oscillation  of  a  pendulum  depends  on  the  in- 
tensity of  the  force  acting  on  the  bob  and  on  the  distance 
from  the  point  of  support  to  the  bob.  It  is  shown  in  ana- 
lytic mechanics  that  the  formula  for  a  complete  oscillation  is 

<  =  27rV|7^, 

where  t  is  the  time,  x  =  3.1416,  I  is  the  length  of  the  pen- 
dulum, and  g  is  the  resultant  acceleration '  produced  by  all 
the  forces  to  which  the  pendulum  is  subject.  If  I  is  deter- 
mined by  measurement  and  t  is  found  by  observations,  the 
resultant  acceleration  is  given  by 

g  = 


Consequently,  the  pendulum  furnishes  a  means  of  finding 
the  gravity  g  at  any  place. 

In  order  to  treat  the  problem  of  determining  the  shape 
of  the  earth  from  a  knowledge  of  g  at  various  places  on  its 
surface,  suppose  first  that  it  is  a  homogeneous  sphere.  If 
this  were  its  shape,  its  attraction  would  be  equal  for  all  points 
on  its  surface.  But  the  gravity  g  would  not  be  the  same 
at  all  places,  because  it  is  the  resultant  of  the  earth's  attrac- 
tion and  the  centrifugal  acceleration  due  to  the  earth's 
rotation.  The  gravity  g  would  be  the  greatest  at  the  poles, 
where  there  is  no  centrifugal  acceleration,  and  least  at  the 
equator,  where  the  attraction  is  exactly  opposed  by  the 
centrifugal  acceleration.  Moreover,  the  value  of  g  would 
vary  from  the  poles  to  the  equator  in  a  perfectly  definite 
manner  which  could  easily  be  determined  from  theoretical 
considerations. 

Now  suppose  the  earth  is  oblate.  It  can  be  shown  mathe- 
matically that  the  attraction  of  an  oblate  body  for  a  particle 
at  its  pole  is  greater  than  that  of  a  sphere  of  equal  volume 
and  density  for  a  particle  on  its  surface,  and  that  at  its 
equator  the  attraction  is  less.     Therefore  at  the  pole,  where 

'  Force  equals  mass  times  acceleration.  On  a  large  pendulum  the  force  of 
gravity  is  greater  but  the  acceleration  is  the  same. 


OH.  II,  15]  THE   EARTH  37 

there  is  no  centrifugal  acceleration,  g  is  greater  on  an  oblate 
body  than  it  is  on  an  equal  sphere.  On  the  other  hand,  at 
the  equator  g  is  less  on  the  oblate  body  than  on  the  sphere 
both  because  the  attraction  of  the  former  is  less,  and  also 
because  its  equator  is  farther  from  its  axis  so  that  the  cen- 
trifugal acceleration  is  greater.  That  is,  the  manner  in 
which  g  varies  from  pole  to  equator  depends  upon  the  oblate- 
ness  of  the  earth,  and  it  can  be  computed  when  the  oblate- 
ness  is  given.  Conversely,  when  g  has  been  foimd  by  ex- 
periment, the  shape  of  the  earth  can  be  computed. 

Very  extensive  determinations  of  g  by  means  of  the  pen- 
dulum, taken  in  connection  with  the  mathematical  theory, 
not  only  prove  that  the  earth  is  oblate,  but  give  a  degree  of 
flattening  agreeing  closely  with  that  obtained  from  the 
measurement  of  arcs. 

The  question  arises  why  g  is  determined  by  means  of  the 
pendulum.  Its  variations  cannot  be  found  by  using  balance 
scales,  because  the  forces  on  both  the  body  to  be  weighed  and 
the  counter  weights  vary  in  the  same  proportion.  However, 
the  variations  in  g  can  be  determined  with  some  approxima- 
tion by  employing  the  spring  balance.  The  choice  between 
the  spring  balance  and  the  pei^dulum  is  to  be  settled  on  the 
basis  of  convenience  and  accuracy.  It  is  obvious  that  spring 
balances  are  very  convenient,  but  they  are  not  very  accurate. 
On  the  other  hand,  the  pendulum  is  capable  of  furnishing 
the  variation  of  g  with  almost  indefinite  precision  by  the 
period  in  which  it  vibrates.  Suppose  the  pendulum  is 
moved  from  one  place  to  another  where  g  differs  by  one 
hundred-thousandth  of  its  value.  This  small  difference  could 
not  be  detected  by  the  use  of  spring  balances,  however  many 
times  the  attempt  might  be  made.  It  follows  from  the 
formula  that  the  time  of  a  swing  of  the  pendulum  would  be 
changed  by  about  one  two-hundred-thousandth  of  its  value. 
If  the  time  of  a  complete  oscillation  were  a  second,  for  ex- 
ample, the  difference  could  not  be  detected  in  a  second ;  but 
the  deviation  for  the  following  second  would  be  equal  to 


38  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  ii,  15 

that  in  the  first,  and  the  difference  would  be  doubled.  The 
effect  would  accumulate,  second  after  second,  and  in  a  day 
of  86,400  seconds  it  would  amount  to  nearly  one  half  of  a 
second,  a  quantity  which  is  easily  measured.  In  ten  days 
the  difference  would  amount  to  about  4.3  seconds.  The 
important  point  in  the  pendulum  method  is  that  the  effects 
of  the  quantities  to  be  measured  accumulate  until  they  be- 
come observable. 

16.  The  Theoretical  Shape  of  the  Earth.  —  The  oblateness 
of  the  earth  is  not  an  accident ;  its  shape  depends  on  its 
size,  mass,  distribution  of  density,  and  rate  of  rotation.     If 


Fig.   11.  ^  Oblate  spheroid.  Fig.   12.  —  Prolate  spheroid. 

it  were  homogeneous,  its  shape  could  be  theoretically  deter- 
mined without  great  difficulty.  It  has  been  found  from 
mathematical  discussions  that  if  a  homogeneous  fluid  body 
is  slowly  rotating  it  may  have  either  of  two  forms  of  equi- 
librium, one  of  which  is  nearly  spherical  while  the  other  is 
very  much  flattened  like  a  discus.  These  figures  are  not 
simply  oblate,  but  they  are  figures  known  as  spheroids.  A 
spheroid  is  a  solid  generated  by  the  rotation  of  an  ellipse 
(Art.  53)  about  one  of  its  diameters.  Figure  11  is  an  oblate 
spheroid  generated  by  the  rotation  of  the  ellipse  PQP'Q' 
about  its  shortest  diameter  PP'.  Its  equator  is  its  largest 
circumference.  Figure  12  is  a  prolate  spheroid  generated 
by  the  rotation. of  the  ellipse  PQP'Q'  about  its  longest  diam- 
eter PP'.  The  equator  of  this  figure  is  its  smallest  cir- 
cumference. The  oblate  and  prolate  spheroids  are  funda- 
mentally different  in  shape. 


CH.  II,  17]  THE   EARTH  39 

Of  the  two  oblate  spheroids  which  theory  shows  are 
figures  of  equilibrium  for  slow  rotation,  that  which  is  the 
more  nearly  spherical  is  stable,  while  the  other  is  unstable. 
That  is,  if  the  former  were  disturbed  a  little,  it  would 
retake  its  spheroidal  form,  while  if  the  latter  were  deformed 
a  Uttle,  it  would  take  an  entirely  different  shape,  or  might 
even  break  all  to  pieces.  In  spite  of  the  fact  that  the  earth 
is  neither  a  fluid  nor  homogeneous,  its  shape  is  ahnost 
exactly  that  of  the  more  nearly  spherical  oblate  spheroid 
corresponding  to  its  density  and  rate  of  rotation.  This  fact 
might  tempt  one  to  the  conclusion  that  it  was  formerly  in  a 
fluid  state.  But  this  conclusion  is  not  necessarily  sound, 
because,  in  such  an  enormous  body,  the  strains  which  would 
result  from  appreciable  departure  from  the  figure  of  equi- 
librium would  be  so  great  that  they  could  not  be  withstood 
by  the  strongest  material  known.  Besides  this,  if  the  con- 
ditions for  equilibrium  were  not  exactly  satisfied  by  the 
solid  parts  of  the  earth,  the  water  and  atmosphere  would 
move  and  make  compensation. 

The  sun,  moon,  and  planets  are  bodies  whose  forms  can 
Hkewise  be  compared  with  the  results  furnished  by  theory. 
Their  figures  agree  closely  with  the  theoretical  forms.  The 
only  appreciable  disagreements  are  in  the  case  of  Jupiter 
and  Saturn,  both  of  which  are  more  nearly  spherical  than 
the  corresponding  homogeneous  bodies  would  be.  The 
reason  for  this  is  that  these  planets  are  very  rare  in  their 
outer  parts  and  relatively  dense  at  their  centers.  It  is 
probable  that  they  are  even  more  stable  than  the  correspond- 
ing homogeneous  figures. 

17.  Different  Kinds  of  Latitude.  —  It  was  seen  in  Art. 
12  that  perpendiculars  to  the  water-level  surface  of  the 
earth,  except  on  the  equator  and  at  the  poles,  do  not  pass 
through  the  center  of  the  earth.  This  leads  to  the  defini- 
tion of  different  kinds  of  latitude. 

The  geometrically  simplest  latitude  is  that  defined  by  a 
line  from  the  center  of  the  earth  to  the  point  on  its  surface 


40 


AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  17 


occupied  by  the  observer.  Thus,  in  Fig.  13,  PC  is  the  earth's 
axis  of  rotation,  QC  is  in  the  plane  of  its  equator,  and  0  is 
the  position  of  the  observer.  The  angle  I  is  called  the  geo- 
centric latitude. 

The  observer  at  0  cannot  see  the  center  of  the  earth  and 
cannot  locate  it  by  any  kind  of  observation  made  at  his 
station  alone.    Consequently,  he  cannot  directly  determine  I. 

All  he  has  is  the  perpen^ 
dicular  to  the  surface  de- 
fined by  his  plumb  Une 
which  strikes  the  line  CQ 
at  A.  The  angle  Ix  be- 
tween this  line  and  CQ  is 
his  astronomical  latitude. 
The  difference  between 
the  geocentric  and  astro- 
nomical latitudes  varies 
from  zero  at  the  poles 
and  equator  to  about  11' 
in  latitude  45°. 
Sometimes  the  plumb  line  has  an  abnormal  direction 
because  of  the  attractions  of  neighboring  mountains,  or 
because  of  local  excesses  or  deficiencies  of  matter  under  the 
surface.  The  astronomical  latitude,  when  corrected  for  these 
anomalies,  is  called  the  geographical  latitude.  The  astro- 
nomical and  geographical  latitudes  rarely  differ  by  more  than 
a  few  seconds  of  arc. 

18.  Historical  Sketch  of  Measurements  of  the  Earth.  — 
While  the  earth  was  generally  supposed  to  be  flat  down  to 
the  time  of  Columbus,  yet  there  were  several  Greek  philoso- 
phers who  believed  that  it  was  a  sphere.  The  earliest  phi- 
losopher who  is  known  certainly  to  have  mai'ntained  that 
the  earth  is  spherical  was  Pythagoras,  author  -of  the  famous 
Pythagorean  proposition  of  geometry,  who  lived  from  about 
569  to  490  B.C.  He  was  followed  in  this  conclusion,  among 
others,  by  Eudoxus  (407-356  b.c),  by  Aristotle  (384-322 


Fig.  13. 


-Geocentric  and  astronomical 
latitudes. 


CH.  It,  18]  THE   EARTH  41 

B.C.),  the  most  famous  philosopher  of  antiquity  if  not  of  all 
time,  and  by  Aristarchus  of  Samos  (310-250  B.C.).  But 
none  of  these  men  seems  to  have  had  so  clear  convictions  as 
Eratosthenes  (275-194  b.c),  who  not  only  believed  in  the 
earth's  sphericity  but  undertook  to  determine  its  dimensions. 
He  had  noticed  that  the  altitude  of  the  pole  star  was  less 
when  he  was  in  Egypt  than  when  he  was  farther  north  in 
Greece,  and  he  correctly  interpreted  this  as  meaning  that 
in  traveling  northward  he  journeyed  around  the  curved  sur- 
faqe  of  the  earth.  By  very  crude  means  he  undertook  to 
measure  the  length  of  a  degree  in  Egypt,  and  in  spite  of  the 
fact  that  he  had  neither  acburate  instruments  for  obtaining 
the  distances  on  the  surface  of  the  earth,  nor  telescopes 
with  which  to  determine  the  changes  of  the  direction  of 
the  plumb  line  with  respect  to  the  stars,  he  secured  results 
that  were  not  surpassed  in  accuracy  until  less  than  300 
years  ago. 

After  the  dechne  of  the  Greek  civilization  and  science,  no 
progress  was  made  in  proving  the  earth  is  spherical  until  the 
voyage  of  Columbus  in  1492.  His  ideas  regarding  the  size 
of  the  earth  were  very  erroneous,  as  is  shown  by  the  fact 
that  he  supposed  lie  had  reached  India  by  crossing  the  Atlan- 
tic Ocean.  The  great  explorations  and  geographical  dis- 
coveries that  quickly  followed  the  voyages  of  Columbus  con- 
vinced men  that  the  earth  is  at  least  globular  and  gave  them 
some  idea  of  its  dimensions. 

There  were  no  serious  attempts  made  to  obtain  accurate 
knowledge  of  the  shape  and  size  of  the  earth  until  about  the 
middle  of  the  seventeenth  century.  The  first  results  of  any 
considerable  degree  of  accuracy  were  obtained  in  1671  by 
Picard  from  a  measurement  of  an  arc  in  France. 

In  spite  of  the  fact  that  Newton  proved  in  1686  that  the 
earth  is  oblate,  the  conclusion  was  by  no  means  universally 
accepted.  Imperfections  in  the  measures  of  the  French  led 
Cassini  to  maintain  until  about  1745  that  the  earth  is  pro- 
late.   But  the  French  were,  taking  hold  of  the  question  in 


42  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  ii,  18 

earnest  and  they  finally  agreed  with  the  conclusion  of  New- 
ton. They  extended  the  arc  that  Picard  had  started  from 
the  Pyrenees  to  Dunkirk,  an  angular  distance  of  9°.  The 
results  were  pubhshed  in  1720.  They  sent  an  expedition  to 
Peru,  on  the  equator,  in  1735,  under  Bouguer,  Condamine, 
and  Godin.  By  1745  these  men  had  measured  an  arc  of  3°. 
In  the  meantime  an  expedition  to  Lapland,  near  the  Arctic 
circle,  had  measured  an  arc  of  1°.  On  comparing  these 
measurements  it  was  found  that  a  degree  of  latitude  is 
greater  the  farther  it  is  from  the  equator. 

In  the  last  century  all  the  principal  governments  of  the 
world  have  carried  out  very  extensive  and  accurate  surveys 
of  their  possessions.  The  English  have  not  only  triangulated 
the  British  Isles  but  they  have  done  an  enormous  amount  of 
work  in  India  and  Africa.  The  Coast  and  Geodetic  Survey 
in  the  United  States  has  triangulated  with  unsurpassed  pre- 
cision a  great  part  of  the  country.  They  have  run  a  level 
from  the  Atlantic  Ocean  to  the  Pacific.  The  names  most 
often  encountered  in  this  connection  are  Clarke  of  England, 
Helmert  of  Germany,  and  Hayford  of  the  United  States. 
Hayford  has  taken  up  an  idea  first  thrown  out  by  the  Eng- 
lish in  connection  with  their  work  in  India  along  the  borders 
of  the  Himalaya  Mountains, .  and  by  using  an  enormous 
amount  of  observational  data  and  making  appalling  com- 
putations he  has  placed  it  on  a  firm  basis.  The  observations 
in  India  showed  that  under  the  Himalaya  Mountains  the 
earth  is  not  so  dense  as  it  is  under  the  plains  to  the  south. 
Hayford  has  proved  that  the  corresponding  thing  is  true  in 
the  United  States,  even  in  the  case  of  very  moderate  eleva- 
tions and  depressions.  Moreover,  deficiency  in  density 
under  the  elevated  places  is  just  enough  to  offset  the  eleva- 
tions, so  that  the  total  weight  of  the  material  along  every 
radius  from  the  surface  of  the  earth  to  its  center  is  almost 
exactly  the  same.  This  theory  is  known  as  the  theory  of 
isostasy,  and  the  earth  is  said  to  be  in  almost  perfect  iso- 
static  adjustment. 


CH.  ii.aQ]  THE   EARTH  43 

II.  QUESTIONS 

1.  In  order  to  prove  the  sphericity  of  the  earth  by  measurement 
of  arcs,  would  it  be  sufflcient  to  measure  only  along  meridians? 
(Consider  the  anchor  ring.)  * 

2.  Do  the  errors  in  triangulation  accumulate  with  the  length  of 
the  distance  measured  ?  Do  tjie  errors  in  the  astronomical  deter- 
mination of  the  angular  length  of  the  arc  increase  with  its  length  ? 

3.  How  accurately  must  a  base  line  of  five  miles  be  measured  in 
order  that  it  may  not  introduce  an  error  in  the  determination  of  the 
earth's  circumference  of  more  than  1000  feet  ? 

4.  Which  of  the  reasons  given  in  Art.  11  actually  prove,  so  far 
as  they  go,  that  the  earth  is  spherical?  What  other  reasons  are 
there  for  believing  it  is  spherical  ? 

5.  The  acceleration  g  in  mid-latitudes  is  about  32.2  feet  per 
second ;  how  long  would  a  pendulum  have  to  be  to  swing  in  1,  2,  3,  4 
seconds  ? 

6.  Draw  to  scale  a  meridian  section  of  a  figure  having  the  earth's 
oblateness. 

7.  Newton  s  proof  of  the  earth's  oblateness  depends  on  the 
knowledge  that  the  earth  rotates ;  what  proofs  of  it  do  not  depend 
upon  this  knowledge  ? 

8.  Suppose  time  can  be  measured  with  an  error  not  exceeding 
one  tenth  of  a  second ;  how  aecm-ately  can  g  be  determined  by  the 
pendulum  in  10  days  ? 

9.  Suppose  the  soMd  part  of  the  earth  were  spherical  and  per- 
fectly rigid ;  what  would  be  the  distribution  of  land  and  water  over 
the  surface  ? 

10.  Is  the  astronomical  latitude  greater  than,  or  equal  to,  the 
geocentric  latitude  for  all  points  on  the  earth's  surface  ? 

11.  What  distance  on  the  earth's  surface  corresponds  to  a  degree 
of  arc,  a  minute  of  are,  a  second  of  are  ? 

12.  Which  of  the  proofs  of  the  earth's  sphericity  depend  upon 
modern  discoveries  and  measurements  ? 

II.  The  Mass  of  the  Earth  and  the  Condition  of 
ITS  Interior 
19.  The  Principle  by  which  Mass  is  Determined.  —  It  is 
important  to  understand  clearly  the  principles  which  are  at 
the  foundation  of  any  subject  in  which  one  may  be  interested, 
and  this  appUes  in  the  present  problem.  The  ordinary 
method  of  determining  the  mass  of  a  body  is  to  weigh  it. 


44  AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ii,  19 

That  is  the  way  in  which  the  quantity  of  most  commodities, 
such  as  coal  or  ice  or  sugar,  is  found.  The  reason  a  body 
has  weight  at  the  surface  of  the  earth  is  that  the  earth 
attracts  it.  It  will  be  seen  later  (Art.  40)  that  the  body 
attracts  the  earth  equally  in  the  opfJosite  direction.  Con- 
sequently, the  real  property  of  a  body  by  which  its  inass  is 
determined  is  its  attraction  for  some  other  body.  The 
underlying  principle  is  that  the  mass  of  a  body  is  proportional 
to  the  attraction  which  it  has  for  another  body. 

Now  consider  the  problem  of  finding  the  mass  of  the 
earth,  which  must  be  solved  by  considering  its  attraction 
for  some  other  body.  Its  attraction  for  any  given  mass,  for 
example,  a  cubic  inch  of  iron,  can  easily  be  measured.  But 
this  does  not  give  the  mass  of  the  earth  compared  to  the 
cubic  inch  of  iron.  It  is  necessary  to  compare  the  attrac- 
tion of  the  earth  for  the  iron  with  the  attraction  of  some  other 
fully  known  body,  as  a  lead  ball  of  given  size,  for  the  same 
unit  of  iron.  Since  the  amount  of  attraction  of  one  body 
for  another  depends  upon  their  distance  apart,  it  is  neces- 
sary to  measure  the  distance  from  the  lead  ball  to  the  at- 
tracted body,  and  also  to  know  the  distance  of  the  attracted 
body  from  the  center  of  the  earth.  For  this  reason  the  mass 
of  the  earth  could  not  be  found  until  after  its  dimensions 
had  been  ascertained.  By  comparing  the  attractions  of  the 
earth  and  the  lead  ball  for  the  attracted  body,  and  making 
proper  adjustments  for  the  distances  of  their  respective 
centers  from  it,  the  number  of  times  the  earth  exceeds  the 
lead  ball  in  mass  can  be  determined. 

Not  only  is  the  mass  of  the  earth  computed  from  its  at- 
traction, but  the  same  principle  is  the  basis  for  determining 
the  mass  of  every  other  celestial  body.  The  masses  of 
those  planets  that  have  satellites  are  easily  found  from  their 
attractions  for  their  respective  satelUtes,  and  when  two 
stars  revolve  around  each  other  in  known  orbits  their  masses 
are  defined  by  their  mutual  attractions.  There  is  no  means 
of  determining  the  mass  of  a  single  star. 


CH.  11,  20]  THE  EARTH  45 

20.  The  Mass  and  Density  of  the  Earth.  —  By  applica- 
tions (Arts.  21,  22)  of  the  principle  in  Art.  19  the  mass  of 
the  earth  has  been  found.  If  it  were  weighed  a  small 
quantity  at  a  time  at  the  surface,  its  total  weight  in  tons 
would  be  6  X  W^,  or  6  followed  by  21  ciphers.  This 
makes  no  appeal  to  the  imagination  because  the  numbers 
are  so  extremely  far  beyond  all  experience.  A  much  btetter 
method  is  to  give  its  density,  which  is  obtained  by  divid- 
ing its  mass  by  its  volume.  With  water  at  its  greatest 
density  as  a  standard,  the  average  density  of  the  earth 
^is  5.53. 

The  average  density  of  the  earth  to  the  depth  of  a  mile 
or  two  is  in  the  neighborhood  of  2.75.  Therefore  there  are 
much  denser  materials  in  the  earth's  interior ;  their  greater 
density  may  be  due  either  to  their  composition  or  to  the 
great  pressure  to  which  they  are  subject.  The  density  of 
quartz  (sand)  is  2.75,  Umestone  3.2,  cast  iron  7.1,  steel  7.8, 
lead  11.3,  mercury  13.6,  gold  19.3,  and  platinum  21.5.  It 
follows  that  no  considerable  part  of  the  earth  can  be  com- 
posed of  such  heavy  substances  as  mercury,  gold,  and  plati- 
num, but,  so  far  as  these  considerations  bear  on  the  question, 
it  might  be  largely  iron. 

The  distribution  of  density  in  the  earth  was  worked  out 
over  100  years  ago  by  Laplace  on  the  basis  of  a  certain  as- 
sumption regarding  the  compressibility  of  the  matter  of 
which  it  is  composed.  The  results  of  this  computation 
have  been  compared  with  all  the  phenomena  on  which  the 
disposition  of  the  mass  of  the  earth  has  an  influence,  and  the 
results  have  been  very  satisfactory.  Hence,  it  is  supposed 
that  this  law  represents  approximately  the  way  the  density 
of  the  earth  increases  from  its  surface  to  its  center.  Accord- 
ing to  this  law,  taking  the  density  of  the  surface  as  2.72, 
the  densities  at  depths  of  1000,  2000,  3000  miles,  and  the 
center  of  the  earth  are  respectively  5.62,  8.30,  10.19,  10.87. 
At  no  depth  is  the  average  density  so  great  as  that  of  the 
heavier  metals. 


46 


AN   INTRODUCTION   TO   ASTRONOMY      [ch.  ii,  21 


21.  Determination  of  the  Density  of  the  Earth  by  Means 
of  the  Torsion  Balance.  —  The  whole  difficulty  in  deter- 
mining the  density  of  the  earth  is  due  to  the  fact  that  the 
attractions  of  masses  of  moderate  dimensions  are  so  feeble 
that  they  almost  escape  detection  with  the  most  sensitive 
apparatus.  The  problem  from  an  experimental  point  of" 
view  reduces  to  that  of  devising  a  means  of  measuring  ex- 
tremely minute  forces.  It  has  been  solved  most  successfully 
by  the  torsion  balance. 

The  torsion  balance  consists  essentially  of  two  small  balls, 
bb  in  Fig.  14,  connected  by  a  rod  which  is  suspended  from 


Fig.  14.  —  The  torsion  balance. 

the  point  0  by  a  quartz  fiber  OA.  If  the  apparatus  is  left 
for  a  considerable  time  in  a  sealed  case  so  that  it  is  not  dis- 
turbed by  air  currents,  it  comes  to  rest.  Suppose  the  balls 
bb  are  at  rest  and  that  the  large  balls  BB  are  carefully 
brought  near  them  on  opposite  sides  of  the  connecting  rod, 
as  shown  in  the  figure.  They  exert  sUght  attractions  for  the 
small  balls  and  gradually  move  them  against  the  feeble 
resistance  of  the  quartz  fiber  to  torsion  (twisting)  to  the 
pbsition  b'b".  The  resistance  of  the  quartz  fiber  becomes 
greater  the  more  it  is  twisted,  and  finally  exactly  balances 
the  attraction  of  the  large  balls.  The  forces  involved  are  so 
small  that  several  hours  may  be  required  for  the  balls  to 
reach  their  final  positions  of  rest.  But  they  will  finally  be 
reached  and  the  angle  through  which  the  rod  has  been  turned 
can  be  recorded. 


CH.  II,  21]  THE   EAKTH  47 

The  next  problem  is  to  determine  from  the  deflection 
which  the  large  balls  have  produced  how  great  the  force  is 
which  they  have  exerted.  This  would  be  a  simple  matter  if 
it  were  known  how  much  resistance  the  quartz  fiber  offers 
to  twisting,  but  the  resistance  is  so  exceedingly  small  that 
it  cannot  be  directly  determined.  However,  it  can  be  found 
by  a  very  interesting  indirect  method. 

Suppose  the  large  balls  are  removed  and  that  the  rod 
connecting  the  small  balls  is  twisted  a  httle  out  of  its  posi- 
tion of  equihbrium.  It  will  then  turn  back  because  of  the 
resistance  offered  to  twisting  by  the  quartz  fiber,  and  will 
rotate  past  the  position  of  equilibrium  almost  as  far  as  it 
was  originally  displaced  in  the  opposite  direction.  Then 
it  will  return  and  vibrate  back  and  forth  until  friction  de- 
stroys its  motion.  It  is  evident  that  the  characteristics  of 
the  oscillations  are  much  like  those  of  a  vibrating  pendulum. 
The  formula  connecting  the  various  quantities  involved  is 

t  =  2  xVi/f, 

where  t  is  the  time  of  a  complete  oscillation  of  the  rod 
joining  h  and  h,  I  is  the  distance  from  A  to  h,  and  /  is  the 
resistance  of  torsion.  This  equation  differs  from  that  for 
the  pendulum.  Art.  15,  only  in  that  g  has  been  replaced  by/. 
Now  I  is  measured,  t  is  observed,  and  /  is  computed  from  the 
equation  with  great  exactness  however  small  it  may  be. 

Now  that  /  and  g  are  known  it  is  easy  to  compute  the 
mass  of  the  earth  by  means  of  the  law  of  gravitation  (Art. 
146).  Let  E  represent  the  mass  of  the  earth,  R  its  radius, 
2  B  the  mass  of  the  two  large  balls,  and  r  the  distances  from 
BB  to  66  respectively.  Then,  since  gravitation  is  propor- 
tional to  the  attracting  mass  and  inversely  as  the  square  of 
its  distance  from  the  attracted  body,  it  follows  that 
E^  2B  ^ 

In  this  proportion  the  only  unknown  is  E,  which  can  there- 
fore be  computed. 


48 


AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  22 


22.  Determination  of  the  Density  of  the  Earth  by  the 
Mountain  Method.  —  The  characteristic  of  the  torsion 
balance  is  that  it  is  very  delicate  and  adapted  to  measuring 
very  small  forces ;  the  characteristic  of  the  mountain  method 
is  that  a  very  large  mass  is.  employed,  and  the  forces  are 
larger.  In  the  torsion  balance  the  balls  BB  are  brought 
near  those  suspended  by  the  quartz  fiber  and  are  removted 
at  will.  A  mountain  cannot  be  moved,  and  the  advantage 
of  using  a  large  mass  is  at  least  partly  coimterbalanced  by 
this  disadvantage.     The  necessity  for  moving  the  attracting 

body  (in  this  case 
^'    ^    A  the    moimtain)    is 

obviated  in  a  very 
ingenious  manner. 

For  simpUcity  let 
the  oblateness  of 
the  earth  be  neg- 
lected in  explaining 
the  mountain 
method.  In  Fig. 
15,  C  is  the  center 
of  the  earth,  M  is 
the  mountain,  and 
Oi  and  O2  are  two 
stations  on  opposite 
sides  of  the  moun- 
tain at  which  plmnb 
lines  are  suspended. 
If  it  were  not  for 
the  attraction  of  the 
mountain  they  would  hang  in  the  directions  OiC  and  O2C. 
The  angle  between  these  fines  at  C  depends  upon  the  distance 
between  the  stations  Oi  and  Oi.  The  distance  between  these 
stations,  even  though  they  are  on  opposite  sides  of  the  moun- 
tain, can  be  obtained  by  triangulation.  Then,  since  the  size 
of  the  earth  is  known,  the  angle  at  C  can  be  computed. 


Fig.  15. 


-  The  mountain  method  of  determining 
the  mass  of  the  earth. 


CH.  II,  22]  THE   EARTH  49 

But  the  attraction  of  the  mountain  for  the  plumb  bobs 
causes  tHe  plumb  lines  to  hang  in  the  directions  OiA  and 
OiA.  The  directions  of  these  lines  with  respect  to  the  stars 
can  easily  be  determined  by  observations,  and  the  difference 
in  their  directions  as  thus  determined  is  the  angle  at  A. 

What  is  desired  is  the  deflections  of  the  plumb  line  pro- 
duced by  the  attractions  of  the  mountain.  It  follows  from 
elementary  geometry  that  the  sum  of  the  two  small  deflec- 
tions COiA  and  CO2A  equals  the  angle  A  minus  the  angle 
C.  Suppose,  for  simplicity,  that  the  mountain  is  sym- 
metrical and  that  the  deflections  are  equal.  Then  each  one 
equals  one  half  the  difference  of  the  angles  A  and  C.  There- 
fore the  desired  quantities  have  been  found. 

When  the  deflection  has  been  found  it  is  easy  to  obtain 
the  relation  of  the  force  exerted  by  the  mountain  to  that 
due  to  the  earth.  Let  Fig.  16  represent  the  g, 
plumb  line  on  a  large  scale.  If  it  were  not 
for  the  mountain  it  would  hang  in  the  direc- 
tion OiBi ;  it  actually  hangs  in  the  direction 
OiB'i.  The  earth's  attraction  is  in  the  direc- 
tion OiBi,  and  that  of  the  mountain  is  in  the 
direction  BiB'i.  The  two  forces  are  in  the 
same  ratio  as  QiBi  is  to  BiB\,  for,  by  the  law  p^^  jg  _  ^^^ 
of  the  composition  of  forces,  only  then  would  deflection  of  a 
the  plumb  line  hang  in  the  direction  OiB'i.         ^"" 

The  problem  of  finding  the  mass  of  the  earth  compared 
to  that  of  the  mountain  now  proceeds  just  Uke  that  of  find- 
ing the  mass  of  the  earth  compared  to  the  balls  BB  in  the 
torsion-balance  method.  The  mountain  plays  the  r61e  of 
the  large  balls.  A  mountain  5000  feet  high  and  broad 
would  cause  nearly  800  times  as  much  deflection  as  that 
produced  by  an  iron  ball  a  foot  in  diameter.  The  advantage 
of  the  large  deflection  is  offset  by  not  having  very  accurate 
means  of  measuring  it,  and  also  by  the  fact  that  it  is  neces- 
sary to  determine  the  mass  of  a  more  or  less  irregular  shaped 
mountain  made  up  of  materials  which  may  lack  much-  of 


50  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  22 

being  uniform  in  density.  In  spite  of  these  drawbacks  this 
method  was  the  first  one  to  give  fairly  accurate  results. 

23.  Determination  of  the  Density  of  the  Earth  by  the 
Pendulum  Method.  —  It  was  explained  in  Art.  15  that  the 
pendulum  furnishes  a  very  accurate  means  of  determining 
the  force  of  gra'^ity.  Its  delicacy  arises  from  the  fact  that 
in  using  it  the  effects  of  the  changes  in  the  forces  accumulate 
indefinitely;  no  such  favorable  circumstances  were  present 
in  the  methods  of  the  torsion  balance  and  the  mountain. 

Suppose  a  pendulum  has  been  swung  at  the  surface  of  the 
earth  so  long  that  the  period  of  its  oscillation  has  been  accu- 
rately determined.  Then  suppose  it  is  taken  at  the  same 
place  down  into  a  deep  pit  or  mine.  The  force  to  which  it 
is  subject  will  be  changed  for  three  different  reasons,  (a)  The 
pendulum  will  be  nearer  the  axis  of  rotation  of  the  earth  and 
the  centrifugal  acceleration  to  which  it  is  subject  will  be 
diminished.  The  relative  change  in  gravity  due  to  this 
cause  can  be  accurately  computed  from  the  latitude  of  the 
position  and  the  depth  of  the  pit  or  mine.  (6)  The  pendu- 
lum will  be  nearer  the  center  of  the  earth,  and,  so  far  as  this 
factor  alone  is  concerned,  the  force  to  which  it  is  subject 
will  be  increased.  Moreover,  the  relative  change  due  to 
this  cause  also  can  be  computed,  (c)  The  pendulum  will  be 
below  a  certain  amount  of  material  whose  attraction  will 
now  be  in  the  opposite  direction.  This  cannot  be  computed 
directly  because  the  amount  of  attraction  due  to  a  ton  of 
matter,  for  example,  is  imknown.  This  is  what  is  to  be 
found  out.  But  from  the  time  of  the  oscillation  of  the  pen- 
dulum at  the  bottom  of  the  pit  or  mine  the  whole  force  to 
which  it  is  subject  can  be  computed.  Then,  on  making  cor- 
rection for  the  known  changes  (a)  and  (6),  the  unknown 
change  (c)  can  be  obtained  simply  by  subtraction.  From 
the  amount  of  force  exerted  by  the  known  mass  above  the 
pendulum,  the  density  of  the  earth  can  be  computed  by 
essentially  the  same  process  as  that  employed  in  the  case 
of  the  torsion-balance  method  and  the  mountain  method. 


CH.  II,  24]  THE   EARTH  51 

24.   Temperature  and  Pressure  in  the  Earth's  Interior.  — 

There  are  many  reasons  for  believing  that  the  interior  of  the 
earth  is  very  hot.  For  example,  volcanic  phenomena  prove 
that  at  least  in  many  localities  the  temperature  is  above  the 
melting  point  of  rock  at  a  comparatively  short  distance 
below  the  earth's  surface.  Geysers  and  hot  springs  show 
that  the  interior  of  the  earth  is  hot  at  many  other  places. 
Besides  this,  the  temperature  has  been  found  to  rise  in  deep 
mines  at  the  rate  of  about  one  degree  Fahrenheit  for  a  de- 
scent of  100  feet,  the  amount  depending  somewhat  on  the 
locality. 

Suppose  the  temperature  should  go  on  increasing  at  the 
rate  of  one  degree  for  every  hundred  feet  from  the  surface 
to  the  center  of  the  earth.  At  a  depth  of  ten  miles  it  would 
be  over  500  degrees,  at  100  miles  over  5000  degrees,  at 
1000  miles  over  50,000  degrees,  and  at  the  center  of  the 
earth  over  200,000  degrees.  While  there  is  no  probabiUty 
that  the  rate  of  increase  of  temperature  which  prevails 
near  the  surface  Seeps  up  to  great  depths,  yet  it  is  reason- 
ably certain  that  at  a  depth  of  a  few  hundred  miles  it  is 
several  thousand  degrees.  Since  almost  every  substance 
melts  at  a  temperature  below  5000  degrees,  it  has  been 
supposed  until  recent  times  that  the  interior  of  the  earth, 
below  the  depth  of  100  miles,  is  hquid. 

But  the  great  pressure  to  which  matter  in  the  interior  of 
the  earth  is  subject  is  a  factor  that  caimot  safely  be  neg- 
lected. A  cyhnder  one  inch  in  cross  section  and  1728 
inches,  or  144  feet,  in  height  has  a  volume  of  one  cubic  foot. 
If  it  is  filled  with  water,  the  pressure  on  the  bottom  equals 
the  weight  of  a  cubic  foot  of  water,  or  62.5  pounds.  The 
pressure  per  square  inch  on  the  bottom  of  the  column  144 
feet  high  having  the  density  2.75,  or  that  of  the  earth's 
crust,  is  172  pounds.  The  pressure  per  square  inch  at  the 
depth  of  a  mile  is  6300  pounds,  or  3  tons  in  round  numbers. 
The  pressure  is  approximately  proportional  to  the  depth  for 
a  co-nsiderable  distance.    Therefore,  the  pressure  per  square 


52  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  24 

inch  at  the  depth  of  100  miles  is  approximately  300  tons, 
and  at  1000  miles  it  is  3000  tons.  However,  the  pressure 
is  not  strictly  proportional  to  the  depth,  and  more  refined 
means  must  be  employed  to  find  how  great  it  is  at  the  earth's 
center.  Moreover,  the  pressure  at  great  depths  depends 
upon  the  distribution  of  mass  in  the  earth.  On  the  basis 
of  the  Laplacian  law  of  density,  which  probably  is  a  good 
approximation  to  the  truth,  the  pressure  per  square  inch  at 
the  center  of  the  earth  is  3,000,000  times  the  atmospheric 
pressure  at  the  earth's  surface,  or  22,500  tons. 

It  is  a  familiar  fact  that  pressure  increases  the  boiling 
points  of  hquids.  It  has  beeh  found  recently  by  experiment 
that  pressure  increases  the  melting  points  of  solids.  There- 
fore, in  view  of  the  enormous  pressures  at  moderate  depths 
in  the  earth,  it  is  not  safe  to  conclude  that  its  interior  is 
molten  without  further  evidence.  The  question  cannot  be 
answered  directly  because,  in  the  first  place,  there  is  no  very 
exact  means  of  determining  the  temperature,  and,  in  the 
second  place,  it  is  not  possible  to  make  experiments  at  such 
high  pressures.  There  are,  however,  several  methods  of 
proving  that  the  earth  is  solid  through  and  through,  and 
they  will  now  be  considered. 

25.  Proof  of  the  Rigidity  and  Elasticity  of  the  Earth  by 
the  Tide  Experiment.  —  Among  the  several  Hnes  of  attack 
that  have  been  made  on  the  question  of  the  rigidity  of  the 
earth,  the  one  depending  on  the  tides  generated  in  the  earth 
by  the  moon  and  sun  has  been  most  satisfactory ;  and  of  the 
methods  of  this  class,  that  devised  by  Michelson  and  carried 
out  in  collaboration  with  Gale,  in  1913,  has  given  by  far 
the  most  exact  results.  Besides,  it  has  settled  one  very 
important  question,  which  no  other  method  has  been  able 
to  answer,  namely,  that  the  earth  is  highly  elastic  instead  of 
being  viscous.  For  these  reasons  the  work  of  Michelson 
and  Gale  will  be  treated  first. 

The  important  difference  between  a  solid  and  a  liquid  is 
that  the  former  offers  resistance  to  deforming  forces  while 


CH.  II,  25]  THE   EARTH  53 

the  latter  does-  not.  If  a  perfect  solid  existed,  no  force  what- 
ever could  deform  it ;  if  a  perfect  liquid  existed,  the  only  re- 
sistance it  would  offer  to  deformation  would  be  the  inertia 
of  the  parts  moved.  Neither  perfect  sohds  nor  absolutely 
perfect  hquids  are  known.  If  a  solid  body  has  the  property 
of  being  deformed  more  and  more  by  a  continually  appUed 
force,  and  if,  on  the  appHcation  of  the  force  being  discon- 
tinued, the  body  not  only  does  not  retake  its  original  form 
but  does  not  even  tend  toward  it,  then  it  is  said  to  be  viscous. 
Putty  is  a  good  example  of  a  material  that  is  viscous.  On 
the  other  hand,  if  on  the  application  of  a  continuous  force 
the  body  is  deformed  to  a  certain  extent  beyond  which  it 
does  not  go,  and  if,  on  the  removal  of  the  force,  it  returns 
absolutely  to  its  original  condition,  it  is  said  to  be  elastic. 
While  there  are  no  soUd  bodies  which  are  either  perfectly 
viscous  or  perfectly  elastic,  the  distinction  is  a  clear  and 
important  one,  and  the  characteristics  of  a  solid  may  be 
described  by  stating  how  far  it  approaches  one  or  the  other 
of  these  ideal  states. 

In  order  to  find  how  the  earth  is  deformed  by  forces  it  is 
,  necessary  to  consider  what  forces  there  are  acting  on  it. 
The  most  obvious  ones  are  the  attractions  of  the  sun  and 
moon.  But  it  is  not  clear  in  the  first  place  that  these  at- 
tractions tend  to  deform  the  earth,  and  in  the  second  place 
that,  even  if  they  have  such  a  tendency,  the  result  is  at 
all  appreciable.  A  ball  of  iron  attracted  by  a  magnet  is  not 
sensibly  deformed,  and  it  seems  that  the  earth  should  be- 
have similarly.  But  the  earth  is  so  large  that  one's  intui- 
tions utterly  fail  in  such  considerations.  The  sun  and 
moon  actually  do  tend  to  alter  the  shape  of  the  earth,  and 
the  amount  of  its  deformation  due  to  their  attractions  is 
measurable.  The  forces  are  precisely  those  that  produce 
the  tides  in  the  ocean. 

It  will  be  sufficient  at  present  to  give  a  rough  idea,  cor- 
rect so  far  as  it  goes,  of  the  reason  that  the  moon  and  sun 
raise  tides  in  the  earth,  reserving  for  Arts.  263,  264  a  more 


54  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  25 

complete  treatment  of  the  question.  In  Fig;  17  let  E  rep- 
resent the  center  of  the  earth,  the  arrow  the  direction  toward 
the  moon,  and  A  and  B  the  points  where  the  line  from  E  to 
the  moon  pierces  the  earth's  surface.  The  moon  is  4000 
miles  nearer  to  A  than  it  is  to  E,  and  4000  miles  nearer  to 
E  than  it  is  to  S.    Therefore  the  attraction  of  the  moon  for 


Fig.   17.  —  The  tidal  bulges  at  A  and  B  on  the  earth  produced  by 
the  moon. 

a  unit  mass  at  A  is  greater  than  it  is  for  a  unit  mass  at  E, 
and  greater  for  a  unit  mass  at  E  than  it  is  for  one  at  B. 
Since  the  distance  from  the  earth  to  the  moon  is  240,000 
miles,  the  distance  of  the  moon  from  A  is  fifty-nine  sixtieths 
of  its  distance  from  E.  Since  the  attraction  varies  inversely 
as  the  square  of  the  distance,  the  force  on  A  is  about  one 
thirtieth  greater  than  that  on  E,  and  the  difference  between 
the  forces  on  E  and  B  is  only  slightly  less. 

It  follows  from  the  relation  of  the  attraction  of  the  moon 
for  masses  at  A,  E,  and  B  that  it  tends  to  pull  the  nearer 
material  at  A  away  from  the  center  of  the  earth  E,  and  the 
center  of  the  earth  away  from  the  more  remote  material  at  B. 
Since  the  forces  are  known,  it  is  possible  to  compute  the 
elongation  the  earth  would  suffer  if  it  were  a  perfect  fluid. 
The  result  is  two  elevations,  or  tidal  bulges,  at  A  and  B. 


CH.  II,  25]  THE   EARTH  56 

The  concentric  lines  shown  in  Fig.  17  are  the  lines  of  equal 
elevation.  A  rather  difficult  mathematical  discussion  shows 
I  that  the  radii  EA  and  EB  would  each  be  lengthened  by 
about  four  feet.  Since  the  earth  possesses  at  least  some 
degree  of  rigidity  its  actual  tidal  elongation  is  somewhat  less 
than  four  feet.  When  it  is  remembered  that  the  uncertainty 
in  the  diameter  of  the  earth,  in  spite  of  the  many  years  that 
have  been  devoted  to  determining  it,  is  still  several  hundred 
feet,  the  problem  of  finding  how  much  the  earth's  elonga- 
tion, as  a  consequence  of  the  rapidly  changing  tidal  forces, 
falls  short  of  four  feet  seems  altogether  hopeless  of  solution. 
Nevertheless  the  problem  has  been  solved. 

Suppose  a  pipe  half  filled  with  water  is  fastened  in  a  hori- 
zontal position  to  the  surface  of  the  earth.  The  water  in  the 
pipe  is  subject  to  the  attraction  of  the  moon.  To  fix  the 
ideas,  suppose  the  pipe  lies  in  the  east-and-west  direction 
in  the  same  latitude  as  the  point  A,  Fig.  17.  Suppose,  first, 
that  the  earth  is  absolutely  rigid  so  that  it  is  not  deformed 
by  the  moon,  and  consider  what  happens  to  the  water  in  the 
pipe  as  the  rotation  of  the  earth  carries  it  past  the  point  A. 
When  the  pipe  is  to  the  west  of  A  the  water  rises  in  its 
eastern  end,  and  settles  correspondingly  in  its  western  end, 
because  the  moon  tends  to  make  an  elevation  on  the  earth 
at  A .  When  the  pipe  is  carried  past  A  to  the  east  the  water 
rises  in  its  western  end  and  settles  in  its  eastern  end.  Since 
the  earth  is  not  absolutely  rigid  the  magnitudes  of  the  tides 
under  the  hypothesis  that  it  is  rigid  cannot  be  experimen- 
tally determined ;  but,  since  all  the  forces  that  are  involved 
are  known,  the  heights  the  tides  would  be  on  a  rigid  earth 
can  be  computed. 

Suppose  now  that  the  earth  yields  perfectly  to  the  disturb- 
ing forces  of  the  moon.  Its  surface  is  in  this  case  always 
the  exact  figure  of  equilibrium.  Consider  the  pipe,  which 
is  attached  to  this  surface,  when  it  is  to  the  west  of  A.  The 
water  would  be  high  in  its  eastern  end  if  the  shape  of  the 
surface  of  the  earth  were  unchanged.    But  the  surface  to 


56  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  25 

the  east  of  it  is  elevated  and  the  pipe  is  raised  with  it.  More- 
over, the  elevation  of  the  surface  is,  under  the  present 
hypothesis,  just  that  necessary  for  equiUbrium.  Therefore, 
in  this  case  there  is  no  tide  at  all  with  respect  to  the  pipe. 

The  actual  earth  is  neither  absolutely  rigid  nor  perfectly 
fluid.  Consequently  the  tides  in  the  pipe  will  actually  be 
neither  their  theoretical  maximum  nor  zero.  The  amount 
by  which  they  fall  short  of  the  value  they  would  have  if  the 
earth  were  perfectly  rigid  depends  upon  the  extent  to  which 
it  yields  to  the  moon's  forces,  and  is  a  measure  of  this  yield- 
ing. Therefore  the  problem  of  finding  how  much  the  earth 
is  deformed  by  the  moon  is  reduced  to  computing  how  great 
the  tides  in  the  pipe  would  be  if  the  earth  were  absolutely 
rigid,  and  then  comparing  these  results  with  the  actual  tides 
in  the  pipe  as  determined  by  direct  experiment.  After  the 
amount  the  earth  yields  has  been  determined  in  this  way, 
its  rigidity  can  be  found  by  the  theory  of  the  deformation 
of  solid  bodies. 

In  the  experiment  of  Michelson  and  Gale  two  pipes  were 
used,  one  lying  in  the  plane  of  the  meridian  and  the  other  in 
the  east-and-west  direction.  In  order  to  secure  freedom 
from  vibrations  due  to  trains  and  heavy  wagons  they  were 
placed  on  the  grounds  of  the  Yerkes  Observatory,  and  to 
avoid  variations  in  temperature  they  were  buried  a  number 
of  feet  in  the  ground.  Since  the  tidal  forces  are  very  small, 
pipes  500  feet  long  were  used,  and  even  then  the  maximum 
tides  were  only  about  two  thousandths  of  an  inch. 

An  ingenious  method  of  measuring  these  small  changes  in 
level  was  devised.  The  ends  of  the  pipes  were  sealed  with 
plane  glass  windows  through  which  their  interiors  could  be 
viewed.  Sharp  pointers,  fastened  to  the  pipe,  were  placed 
just  under  the  surface  of  the  water  near  the  windows.  When 
viewed  from  below  the  level  of  the  water  the  pointer  and  its 
reflected  image  could  be  seen.  Figure  18  shows  an  end  of 
one  of  the  pipes,  S  is  the  surface  of  the  water,  P  is  the  pointer, 
and  P'  is  its  reflected  image.    The  distances  of  P  and  P' 


CH.  II,  26]  THE   EARTH  67 

from  the  surface  iS  are  equal.     Now  suppose  the  water  rises ; 

since  P  and  P'  are  equidistant  from  S,  the  change  in  their 

apparent  distance  is  twice  the  change 

in  the  water  level.     The  distances 

between  P  and  P'  were  accurately     /  ■ 

measured  with  the  help  of  perma-  /  |  , 

nently   fixed    microscopes,  and    the  ' — ^ 

variations  in  the  water  level  were 

determined  within  one  per  cent  of 

their  whole  amount. 

In  order  to  make  clear  the  accuracy  T,      ,„      t,  j    ,    • 

V.J.  ^i°-   18.  — End  of   pipe  in 

oi  the  results,  the  complicated  nature  the  Micheison-Gaie  tide 
of  the  tides  must'  be  pointed  out.  experiment. 
Consider  the  tidal  bulges  A  and  B,  Fig.  17,  which  give  an  idea 
of  what  happened  to  the  water  in  the  pipes.  For  simpKcity, 
fix  the  attention  on  the  east-and-west  pipe,  which  in  the  ex- 
periment was  about  13°  north  of  the  highest  latitude  A  ever 
attains.  The  rotating  earth  carried  it  daily  across  the  merid- 
ian of  A  to  the  north  of  A,  and  similarly  across  the  meridian 
of  B.  When  the  relations  were  as  represented  in  the  dia- 
gram there  were  considerable  tides  in  the  pipe  before  and 
after  it  crossed  the  meridian  at  A  because  it  was,  so  to  speak, 
well  on  the  tidal  bulge.  On  the  other  hand,  when  it  crossed 
the  meridian  of  B  about  12  hours  later,  the  tides  were  very 
small  because  the  bulge  B  was  far  south  of  the  equator. 
But  the  moon  was  not  all  the  time  north  of  the  plane  of  the 
earth's  equator.  Once  each  month  it  was  28°  north  and 
once  each  month  28°  south,  and  it  varied  from  hour  to  hour 
in  a  rather  irregular  manner.  Moreover,  its  distance,  on 
which  the  magnitudes  of  the  tidal  forces  depend,  also  changed 
continuously.  Then  add  to  all  these  complexities  the  cor- 
responding ones  due  to  the  sun,  which  are  unrelated  to  those 
of  the  moon,  and  which  mix  up  with  them  and  make  the 
phenomena  still  more  involved.  Finally,  consider  the  north- 
and-south  pipe  and  notice,  by  the  help  of  Fig.  17,  that  its 
tides  are  altogether  distinct  in  character  from  those  in  the 


58  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  25 

east-and-west  pipe.  With  all  this  in  mind,  remember  that 
the  observations  made  every  two  hours  of  the  day  for  a 
period  of  several  months  agreed  perfectly  in  all  their  char- 
acteristics with  the  results  given  by  theory.  The  only  dif- 
ference was  that  the  observed  tides  were  reduced  in  a  con- 
stant ratio  by  the  yielding  of  the  earth. 

The  perfection  of  this  domain  of  science  is  proved  by  the 
satisfactory  coordination  in  this  experiment  of  a  great  many 
distinct  theories.  The  perfect  agreement  in  their  charac- 
teristics of  more  than  a  thousand  observed  tides  with  their 
computed  values  depended  on  the  correctness  of  the  laws  of 
motion,  the  truth  of  the  law  of  gravitation,  the  size  of  the 
earth,  the  distance  of  the  moon  and  the  theory  of  its  motion, 
the  mass  of  the  moon,  the  distance  to  the  sun  and  the  theory 
of  the  earth's  motion  around  it,  the  mass  of  the  sun,  the 
theory  of  tides,  the  numerous  observations,  and  the  lengthy 
calculations.  How  improbable  that  there  would  be  perfect 
harmony  between  observation  and  theory  in  so  many  cases 
unless  scientific  conclusions  respecting  all  these  things  are 
correct! 

The  extent  to  which  the  earth  yields  to  the  forces  of  the 
moon  was  obtained  from  the  amount  by  which  the  observed 
tides  were  less  than  their  theoretical  values  for  an  unyielding 
earth.  It  was  found  that  in  the  east-and-west  pipe  the  ob- 
served tides  were  about  70  per  cent  of  the  computed,  while 
in  the  north-and-south  pipe  the  observed  tides  were  only 
about  50  per  cent  of  the  computed.  This  led  to  the  astonish- 
ing conclusion,  which,  however,  had  been  reached  earher  by 
Schweydar  on  the  basis  of  much  less  certain  observational 
data,  that  the  earth's  resistance  to  deformation  in  the  east- 
and-west  direction  is  greater  than  it  is  in  the  north-and-south 
direction.  Love  has  suggested  that  the  difference  may  be 
due  indirectly  to  the  effects  of  the  oceanic  tides  on  the  general 
body  of  the  earth. 

On  using  the  amount  of  the  yielding  of  the  earth  estab- 
lished by  observations  and  the  magnitude  of  the  forces  exerted 


CH.  II,  26]  THE   EARTH  59 

by  the  moon  and  sun,  it  was  found  by  the  mathematical 
processes  which  are  necessary  in  treating  such  problems, 
that  the  earth,  taken  as  a  wl)ole,  is  as  rigid  as  steel.  That 
is,  it  resists  deformation  as  much  as  it  would  if  it  were  made 
of  solid  steel  having  throughout  the  properties  of  ordinary 
good  steel. 

The  work  of  Michelson  and  Gale  for  the  first  time  gave  a 
reliable  answer  to  the  question  whether  the  earth  is  viscous 
or  elastic.  It  had  almost  invariably  been  supposed  that 
the  earth  is  viscous,  because  it  was  thought  that  even  if 
the  enormous  pressure  keeps  the  highly  heated  material  of 
its  interior  in  a  solid  state,  yet  it  would  be  only  stiff  like 
a  soHd  is  when  its  temperature  approaches  the  melting  point. 
In  fact,  Sir  George  Darwin  had  built  up  an  elaborate  theory 
of  tidal  evolution  (Arts.  265,  266),  at  the  cost  of  a  number 
of  years  of  work,  on  the  hypothesis  that  the  earth  is  viscous. 
But  the  experiments  of  Michelson  and  Gale  prove  that  it  is 
very  elastic. 

If  the  earth  were  viscous,  it  would  yield  somewhat  slowly 
to  the  forces  of  the  moon  and  sun.  Consequently,  the  tilting 
of  the  surface,  which  carries  the  pipes,  would  lag  behind  the 
forces  which  caused  both  the  tilting  and  the  tides  in  the 
pipes'.  There  is  no  appreciable  lag  of  a  water  tide  in  the 
pipe  only  500  feet  long,  and  consequently  the  observed  and 
computed  tides  would  not  agree  in  phase.  On  the  other 
hand,  if  the  earth  were  elastic,  there  would  be  agreement  in 
phase  between  the  observed  and  computed  tides.  It  is  more 
difficult  practically  to  determine  accurately  the  phase  of  the 
tides  than  it  is  to  measure  their  magnitudes,  but  the  obser- 
vations showed  that  there  is  no  appreciable  difference  in  the 
phases  of  the  observed  and  computed  tides.  These  results 
force  the  conclusion  that  the  elasticity  of  the  earth,  taken  as 
a  whole,  cannot  be  less  than  that  of  steel,  —  a  result  ob- 
viously of  great  interest  to  geologists. 

26.  Other  Proofs  of  the  Earth's  Rigidity.  —  (a)  There  is 
a  method  of  finding  how  much  the  earth  yields  to  the  forces 


60        AN   INTRODUCTION   TO  ASTRONOMY       [ch.  ii,  26 

of  the  moon  and  sun  which  is  fundamentally  equivalent  to 
that  of  measuring  tides  in  a  pipe.  It  depends  on  the  fact 
that  the  position  of  a  pendulum  depends  upon  all  the  forces 
acting  on  it,  and,  if  the  earth  were  in  equihbrium,  the  line 
of  its  direction  would  always  be  perpendicular  to  the  water- 
leyel  surface.  Consequently,  if  the  earth  yielded  perfectly 
to  the  forces  of  the  moon  and  sun,  a  pendulum  would  con- 
stantly remain  perpendicular  to  its  water-level  surface. 
But  if  the  earth  did  not  yield  perfectly,  the  pendulum  would 
undergo  very  minute  oscillations  with  respect  to  the  solid 
part  analogous  to  those  of  the  water  in  the  pipes.  A  modi- 
fication of  the  ordinary  pendulum,  known  as  the  horizontal 
pendulum,  was  found  to  be  sensitive  enough  to  show  the 
oscillations,  giving  the  rigidity  of  the  earth  but  no  satis- 
factory evidence  regarding  its  elasticity. 

(6)  The  principles  at  the  basis  of  the  method  of  employ- 
ing tides  in  pipes  apply  equally  well  to  tides  in  the  ocean. 
Longer  columns  of  water  are  available  in  this  case,  but  there 
is  difficulty  in  obtaining  the  exact  heights  of  the  actual  tides, 
and  very  much  greater  difficulty  in  determining  their  theo- 
retical heights  on  a  shelving  and  irregular  coast  where  they 
would  necessarily  be  observed.  In  fact,  it  has  not  yet  been 
found  possible  to  predict  in  advance  with  any  considerable 
degree  of  accuracy  the  height  of  tides  where  they  have  not 
been  observed.  Yet,  Lord  Kelvin  with  rare  judgment  in- 
ferred on  this  basis  that  the  earth  is  very  rigid. 

(c)  Earthquakes  are  waves  in  the  earth  which  start  from 
some  restricted  region  and  spread  all  over  the  earth,  diminish- 
ing in  intensity  as  they  proceed.  Modem  instruments, 
depending  primarily  on  some  adaptation  of  the  horizontal 
pendulum,  can  detect  important  earthquakes  to  a  distance 
of  thousands  of  miles  from  their  origin.  Earthquake  waves 
are  of  different  types;  some  proceed  through  the  surface 
rocks  around  the  earth  in  undulations  like  the  waves  in  the 
ocean,  while  others,  compressional  in  character  like  waves  of 
sound  in  the  air,  radiate  in  straight  Unes  from  their  sources. 


CH.  II,  26]  THE   EARTH  61 

The  speed  of  a  wave  depends  upon  the  density  and  the 
rigidity  of  the  medium  through  which  it  travels.  This  prin- 
ciple appUes  to  earthquake  wavqs,  and  when  tested  on  those 
which  travel  in  undulations  through  the  surface  rocks  there 
is  good  agreement  between  observation  and  theory.  Con- 
sider its  application  to  the  compressional  waves  that  go 
through  the  earth.  The  time  required  for  them  to  go  from 
the  place  of  their  origin  to  the  place  where  they  are  observed 
is  given  by  the  observations.  The  density  of  the  earth  is 
known.  If  its  rigidity  were  known,  the  time  could  be  com- 
puted ;  but  the  time  being  known,  the  rigidity  can  be  com- 
puted. While  the  results  are  subject  to  some  uncertainties, 
they  agree  with  those  found  by  other  methods. 

(d)  The  attraction  of  the  moon  for  the  equatorial  bulge 
slowly  changes  the  plane  of  the  earth's  equator  (Art.  47). 
The  magnitude  of  the  force  that  causes  this  change  is  known. 
If  the  earth  consisted  of  a  crust  not  more  than  a  few  hundred 
miles  deep  floating  on  a  liquid  interior,  the  forces  would 
cause  the  crust  to  slip  on  the  liquid  core,  just  as  a  vessel  con^ 
taining  water  can  be  rotated  without  rotating  the  water.  If 
the  crust  of  the  earth  alone  were  moved,  it  would  be  shifted 
rapidly  because  the  mass  moved  would  not  be  great.  But 
the  rate  at  which  the  plane  of  the  earth's  equator  is  moved, 
as  given  by  the,  observations,  taken  together  with  the  forces 
involved,  proves  that  the  whole  earth  moves.  When  the 
effects  of  forces  acting  on  such  an  enormous  body  are  con- 
sidered, it  is  found  that  this  fact  means  that  the  earth  has  a 
considerable  degree  of  rigidity. 

(e)  Every  one  knows  that  a  top  may  be  spun  so  that  its 
axis  remains  stationary  in  a  vertical  direction,  or  so  that  it 
wabbles.  Similarly,  a  body  rotating  freely  in  space  may 
rotate  steadily  around  a  fixed  axis,  or  its  axis  of  rotation 
may  wabble.  The  period  of  the  wabbling  depends  upon  the 
size,  shape,  mass,  rate  of  rotation,  and  rigidity  of  the  body. 
In  the  case  of  the  earth  all  these  factors  except  the  last  may 
be  regarded  as  known.     If  it  were  known,  the  rate  of  wab- 


62  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  26 

bling  could  be  computed ;  or,  if  the  rate  of  wabbling  were 
found  from  observation,  the  rigidity  could  be  computed. 
It  has  recently  been  found  that  the  earth's  axis  of  rotation 
wabbles  slightly,  and  the  rate  of  this  motion  proves  that  the 
rigidity  of  the  earth  is  about  that  of  steel. 

27.  Historical  Sketch  on  the  Mass  and  Rigidity  of  the 
Earth.  —  The  history  of  correct  methods  of  attempting  to 
find  the  mass  of  the  earth  necessarily  starts  with  Newton, 
because  the  ideas  respecting  mass  were  not  clearly  formu- 
lated before  his  time,  and  because  the  determination  of  mass 
depends  on  the  law  of  gravitation  which  he  discovered.  By 
some  general  but  inconclusive  reasoning  he  arrived  at  the 
conjecture  that  the  earth  is  five  or  six  timps  as  dense  as 
water. 

The  first  scientific  attempt  to  determine  the  density  of 
the  earth  was  made  by  Maskelyne,  who  used  the  mountain 
method,  in  1774,  in  Scotland.  He  found  4.5  for  the  density 
of  the  earth.  The  torsion  balance,  devised  by  Michell,  was 
first  employed  by  Cavendish,  in  England,  in  1798.  His 
result  agreed  closely  with  those  obtained  by  later  experi- 
menters, among  whom  may  be  mentioned  Baily  (1840)  in 
England,  and  Reich  (1842)  in  Germany,  Cornu  (1872)  in 
France,  Wilsing  (1887)  in  Germany,  Boys  (1893)  in  England, 
and  Braun  (1897)  in  Austria.  The  pendulum  method,  using 
either  a  mountain  or  a  mine  to  secure  difference  in  elevation, 
has  been  employed  a  number  of  times. 

Lord  Kelvin  (then  Sir  William  Thomson)  first  gave  in 
1863  good  reasons  for  beheving  the  earth  is  rigid.  His  con- 
clusion was  based  on  the  height  of  the  oceanic  tides,  as  out- 
fined  in  Art.  26  (b).  The  proof  by  means  of  the  rate  of 
transmissioii  of  earthquake  waves  owes  its  possibifity  largely 
to  John  Milne,  an  EngUshman  who  long  lived  in  Japan, 
which  is  frequently  disturbed  by  earthquakes.  His  interest 
in  the  character  of  earthquakes  stimulated  him  to  the  inven- 
tion of  instruments,  known  as  seismographs,  for  detecting 
and  recording  faint  earth  tremors.     The  change  of  the  posi- 


CH.  II,  27]  THE   EARTH  63 

tion  of  the  plane  of  the  earth's  equator,  known  as  the  pre- 
cession of  the  equinoxes,  has  been  known  observationally 
ever  since  the  days  of  the  ancient  Greeks,  and  its  cause  was 
understood  by  Newton,  but  it  has  not  been  used  to  prove 
the  rigidity  of  the  earth  because  it  takes  place  very  slowly. 
The  wabbling  of  the  axis  of  the  earth  was  first  established 
observationally,  in  1888,  by  Chandler  of  Cambridge,  Mass., 
and  Kiistner  of  Berlin.  The  theoretical  applications  of  the 
rigidity  of  the  earth  were  made  first  by  Newcomb  of  Wash- 
ington, and  then  more  completely  by  S.  S.  Hough  of  Eng- 
land. The  first  attempt  at  the  determination  of  the  rigidity 
of  the  earth  by  the  amount  it  yields  to  the  tidal  forces  of  the 
moon  and  sun  was  made  unsuccessfully  in  1879  by  George 
and  Horace  Darwin,  in  England.  Notable  success  has  been 
achieved  only  in  the  last  15  years,  and  that  by  improvements 
in  the  horizontal  pendulum  and  by  taking  great  care  in 
keeping  the  instnmaents  from  being  disturbed.  The  names 
that. stand  out  are  von  Rebeur-Paschwitz,  Ehlert,  Kortozzi, 
Schweydar,  Hecker,  and  Orloff.  The  observations  of 
Hecker  at  Potsdam,  Germany,  were  especially  good,  and 
Schweydar  made  two  exhaustive  mathematical  discussions 
of  the  subject. 

III.  QUESTIONS 

1.  What  is  the  difference  between  mass  and  weight  ?  Does  the 
weight  of  a  body  depend  on  its  position?  Does  the  inertia  of  a 
body  depend  on  its  position  ? 

2.  Can  the  mass  of  a  small  body  be  determined  from  its  inertia  ? 
Can  the  mass  of  the  earth  be  determined  in  the  same  way  ? 

3.  What  is  the  average  weight  of  a  cubic  mile  of  the  earth  ? 

4.  Discuss  the  relative  advantages  of  the  torsion-balance  method 
and  mountain  method  in  determining  the  density  of  the  earth. 
Which  one  has  the  greater  advantages  ? 

5.  What  is  the  pressure  at  the  bottom  of  an  ocean  six  miles 
deep  ? 

6.  Discuss  the  character  of  the  tides  in  east-and-west  and  north- 
and-south  pipes  during  a  whole  day  when  the  moon  is  in  the  posi- 
tion indicated  in  Fig.  17,  and  when  it  is  over  the  earth's  equator. 


64        AN   INTRODUCTION   TO  ASTRONOMY       [ch.  ii,  27 

7.  What  are  the  advantages  and  disadvantages  of  a  long  pipe 
in  the  tide  experiment  ? 

8.  If  a  body  i»  at  A,  Fig.  17,  is  its  weight  greater  or  less  than 
normal  as  determined  by  spring  balances?  By  balance  scales? 
What  are  the  facts,  if  it  is  at  B? 

9.  Enumerate  the  scientific- theories  and  facts  involved  in  the 
tide  experiment. 

10.  List  the  principles  on  which  the  several  proofs  of  the  earth's 
rigidity  depend.  How  many  fundamentally  different  methods  are 
there  of  determining  its  rigidity? 

III.    The  Eaeth's  Atmosphere 

28.   Composition  and  Mass  of  the  Earth's  Atmosphere.  — 

The  atmosphere  is  the  gaseous  envelope  which  surrounds 
the  earth.  Its  chief  constituents  are  the  elements  nitrogen 
and  oxygen,  but  there  are  also  minute  quantities  of  argon, 
hehum,  neon,  krypton,  xeon,  and  some  other  very  rare  con- 
stituents. When  measured  by  volume  at  the  earth's  sur- 
face, 78  per  cent  of  the  atmosphere  is  nitrogen,  21  per  cent 
is  oxygen,  0.94  per  cent  is  argon,  and  the  remaining  elements, 
occur  in  much  smaller  quantities. 

Nitrogen,  oxygen,  etc.,  are  elements;  that  is,  they  are 
substances  which  are  not  broken  up  into  more  fundamental 
units  by  any  physical  or  any  chemical  changes.  The  thou- 
sands of  different  materials  that  are  found  on  the  earth  are 
all  made  up  of  about  90  elements,  only  about  half  of  which 
are  of  very  frequent  occurrence.  The  union  of  elements 
into  a  chemical  compound  is  a  very  fundamental  matter,  for " 
the  compound  may  have  properties  very  unlike  those  of  any 
of  the  elements  of  which  it  is  composed.  For  example, 
hydrogen,  carbon,  and  nitrogen  are  in  almost  all  food,  but 
hydrocyanic  acid,  which  is  composed  of  these  elements  alone, 
is  a  deadly  poison. 

Besides  the  elements  which  have  been  enumerated,  the 
atmosphere  contains  some  carbon  dioxide,  which  is  a  com- 
pound of  carbon  and  oxygen,  and  water  vapor,  which  is 
a  compound   of  oxygen   and  hydrogen.     In  volume  three 


CH.  II,  29]  THE   EARTH  65 

hundredths  of  one  pfer  cent  of  the  earth's  atmosphere  is 
carbon  dioxide ;  but  this  compound  is  heavier  than  nitrogen 
and  oxygen,  and  by  weight,  0.05  per  cent  of  the  atmosphere 
is  carbon  dioxide.  The  amount  of  water  vapor  in  the  air 
varies  greatly  with  the  position  on  the  earth's  surface  and 
with  the  time.  There  are  also  small  quantities  of  dust,  soot, 
ammonia,  and  many  other  things  which  occur  in  variable 
quantities  and  which  are  considered  as  impurities. 

The  pressure  of  the  atmosphere  at  sea  level  is  about  15 
pounds  per  square  inch  and  its  density  is  about  one  eight- 
hundredth  that  of  water.  This  means  that  the  weight  of  a 
column  of  air  reaching  from  the  earth's  surface  to  the  limits 
of  the  atmosphere  and  having  a  cross  section  of  one  square 
inch  weighs  15  poimds.  The  total  mass  of  the  atmosphere 
can  be  obtained  by  multiplying  the  weight  of  one  column 
by  the  total  area  of  the  earth.  In  this  way  it  is  found 
that  the  mass  of  the  earth's  atmosphere  is  nearly 
6,000,000,000,000,000  tons,  or  approximately  one  miUionth 
the  mass  of  the  sohd  earth.  The  total  mass  of  even  the 
carbon  dioxide  of  the  earth's  atmosphere  is  approximately 
3,000,000,000,000  tons. 

29.  Determination  of  Height  of  Earth's  Atmosphere  from 
Observations  of  Meteors.  —  Meteors,  or  shooting  stars  as 
they  are  commonly  called,  are  minute  bodies,  circulating  in 
interplanetary  space,  which  become  visible  only  when  they 
penetrate  the  earth's  atmosphere  and  are  made  incandescent 
by  the  resistance  which  they  encounter.  The  great  heat 
developed  is  a  consequence  of  their  high  velocities,  which 
ordinarily  are  in  the  neighborhood  of  25  miles  per  second. 

Let  m,  Fig.  19,  represent  the  path  of  a  meteor  before  it 
encounters  the  atmosphere  at  A.  Until  it  reaches  A  it  is 
invisible,  but  at  A  it  begins  to  glow  and  continues  luminous 
until  it  is  entirely  burned  up  at  B.  Suppose  it  is  observed 
from  the  two  stations  Oi  and  O2  which  are  at  a  known  dis- 
tance apart.  The  observations  at  Oi  give  the  angle  AO1O2, 
and  those  at  O2  give  the  angle  AO2O1.     From  these  data  the 


66      AN   INTRODUCTION   TO   ASTRONOMY  [ch.  n,  29 

other  parts  of  the  triangle  can  be  computed  (compare  Art. 

10).  After  the  distance  OiA  has  been  computed  the  perpen- 
dicular height  of 
A  from  the  sur- 
face of  the  earth 
can  be  computed 
by  using  the  angle 
AOiOs.  Similarly, 
the  height  of  B 
above  the  surface 

Fig.  19.  —  Determination  of  the  height  of  meteors.    O^   ^'^^    earth    can 

be  determined. 
Observations  of  meteors  from  two  stations  show  that  they 
ordinarily  become  visible  at  a  height  of  from  60  to  100 
miles.  Therefore,  the  atmosphere  is  sufficiently  dense  to 
a  height  of  about  100  miles  to  offer  sensible  resistance  to 
meteors.  Meteors  usually  disappear  by  the  time  they  have 
descended  to  within  thirty  or  forty  miles  of  the  earth's 
surface. 

30.  Determination  of  Height  of  Earth's  Atmosphere  from 
Observations  of  Aurorse.  —  Aurorae  are  almost  certainly 
electrical  phenomena  of  the  very  rare  upper  atmosphere, 
though  their  nature  is  not  yet  very  well  understood.  Their 
altitude  can  be  computed  from  simultaneous  observations 
made  at  different  stations.  The  method  is  the  same  as  that 
in  obtaining  the  height  of  a  meteor. 

The  southern  ends  of  auroral  streamers  are  usually  more 
than  100  miles  in  height,  and  they  are  sometimes  found  at 
an  altitude  of  500  or  600  miles.  Their  northern  ends  are 
much  lower.  This  means  that  the  density  required  to  make 
meteors  incandescent  is  considerably  greater  than  that  which 
is  sufficient  for  auroral  phenomena. 

31.  Determination  of  Height  of  Earth's  Atmosphere  from 
the  Duration  of  Twilight.  —  Often  after  sunset,  even  to  the 
east  of  the  observer,  high  clouds  are  briUiantly  illuminated 
by  the  rays  of  the  sun  which  still  fall  on  them.     The  higher 


CH.  II,  31] 


THE   EARTH 


67 


the  clouds  are,  the  longer  they  are  illuminated.  Similarly, 
the  sun  shines  on  the  upper  atmosphere  for  a  considerable 
time  after  it  has  set  or  before  it  rises,  and  gives  the  twihght. 
The  duration  of  twilight  depends  upon  the  height  of.  the 
atmosphere.  While  it  is  difficult  to  determine  the  instant 
at  which  the  twihght  ceases  to  be  visible,  observations  show 
that  under  favorable  weather  conditions  it  does  not  disap- 
pear until  the  sun  is  18  degrees  below  the  horizon. 

In  order  to  see  how  the  height  of  the  atmosphere  can  be 
determined  from  the  duration  of  the  twilight,  consider  Fig.  20. 
The  sun's  rays 
come  in  from  the 
left  in  hnes  that 
are  sensibly  par- 
allel. The  ob- 
server at  0  can 
see  the  illumin- 
ated atmosphere 
at  P ;  but  if  the 
atmosphere  were 
much  shallower, 
it  would  not  be 
visible  to  him.  The  region  P  is  midway  between  0  and  the 
sunset  point.  Since  0  is  18  degrees  from  the  sunset  point,  it 
is  possible  to  compute  the  height  of  the  plane  of  the  horizon 
at  P  above  the  surface  of  the  earth.  It  is  found  that  18 
degrees  corresponds  to  an  altitude  of  50  miles.  That  is,  the 
atmosphere  extends  to  a  height  of  50  miles  above  the  earth's 
surface  in  quantities  sufficient  to  produce  twilight. 

The  results  obtained  by  the  various  methods  for  determin- 
ing the  height  of  the  atmosphere  disagree  because  its  density 
decreases  with  altitude,  as  is  found  by  ascending  in  balloons, 
and  different  densities  are  required  to  produce  the  different 
phenomena.  It  will  convey  the  correct  idea  for  most  appli- 
cations to  state  that  the  atmosphere  does  not  extend  in  ap- 
preciable quantities  beyond  100  miles  above  the  earth's  sur- 


FlG. 


20.  —  Determination    of    the    height    of 
atmosphere  from  the  duration  of  twilight. 


the 


68  AN   INTRODUCTION   TO  ASTRONOMY     [ch.  ii,  31 

face.  At  this  altitude  its  density  is  of  the  order  of  one  four- 
millionth  of  that  at  the  surface.  When  the  whole  earth  is 
considered  it  is  found  that  the  atmosphere  forms  a  relatively- 
thin  layer.  If  the  earth  is  represented  by  a  globe  8  inches 
in  diameter,  the  thickness  of  the  atmosphere  on  the  same 
scale  is  only  about  one  tenth  of  an  inch. 

32.  The  Eonetic  Theory  of  Gases.  —  It  has  been  stated 
that  every  known  substance  on  the  earth  is  composed  of 
about  90  fundamental  elements.  A  chemical  combination 
of  atoms  is  called  a  molecule.  A  molecule  of  oxygen  con- 
sists of  two  atoms  of  oxygen,  a  molecule  of  water  of  two 
atoms  of  hydrogen  and  one  of  oxygen,  and  similarly  for  all 
substances.  Some  molecules  contain  only  a  few  atoms  and 
others  a  great  many ;  for  example,  a  molecule  of  cane  sugar 
is  composed  of  12  atoms  of  carbon,  22  of  hydrogen,  and  11 
of  oxygen.  As  a  rule  the  compounds  developed  in  connec- 
tion with  the  life  processes  contain  many  atoms. 

The  molecules  are  all  very  minute,  though  their  dimen- 
sions doubtless  vary  with  the  number  and  kind  of  atoms 
they  contain."  Lord  Kelvin  devised  a  number  of  methods 
of  determining  their  size,  or  at  least  the  distances  between 
their  centers.  In  water,  for  example,  there  are  in  round 
numbers  500,000,000  in  a  Une  of  them  one  inch  long,  or  the 
cube  of  this  number  in  a  cubic  inch. 

In  solids  the  molecules  are  constrained  to  keep  essentially 
the  same  relations  to  one  another,  though  they  are  capable 
of  making  complicated  small  vibrations.  In  liquids  the 
molecules  continually  suffer  restraints  from  neighboring 
molecules,  but  their  relative  positions  are  not  fixed  and  they 
move  around  among  one  another,  though  not  with  perfect 
freedom.  In  gases  the  molecules  are  perfectly  free  from  one 
another  except  when  they  collide.  They  move  with  great 
speed  and  collide  with  extraordinary  frequency ;  but,  in  spite 
of  the  frequency  of  the  collisions,  the  time  during  which 
they  are  uninfluenced  by  their  neighbors  is  very  much  greater 
than  that  in  which  they  are  in  effective  contact. 


CH.  II,  33]  THE   EARTH  69 

The  pressure  exerted  by  a  gas  is  due  to  the  impact  of  its 
molecules  on  the  walls  of  the  retaining  vessel.  To  make  the 
ideas  definite,  consider  a  cubic  foot  of  atmosphere  at  sea-level 
pressure.  Its  weight  is  about  one  and  one  fourth  ounces, 
but  it  exerts  a  pressure  of  15  pounds  on  each  square  inch  of 
each  of  its  six  surfaces,  or  a  total  pressure  on  the  surface  of 
the  cube  of  more  than  six  tons.  This  imphes  that  the  mole- 
cules move  with  enormous  speed.  They  do  not  all  move 
with  the  same  speed,  but  some  travel  slowly  while  others  go 
much  faster  than  the  average.  Theoretically,  at  least,  in 
every  gas  there  are  molecules  moving  with  every  velocity, 
however  great,  but  the  number  of  those  having  any  given 
velocity  diminishes  rapidly  as  its  difference  from  the  aver- 
age velocity  increases.  The  average  velocity  of  molecules 
in  common  air  at  ordinary  temperature  and  pressure  is  more 
than  1600  feet  per  second,  and  on  the  average  each  mole- 
cule has  5,000,000,000  colUsions  per  second.  Therefore  the 
average  distance  traveled  between  coUisions  is  only  about 
asoVoo  of  an  inch. 

From  the  kinetic  theory  of  gases  it  is  possible  to  deter- 
mine how  fast  the  density  of  the  air  diminishes  with  increase 
of  altitude.  It  is  found  that  about  one  half  of  the  earth's 
atmosphere  is  within  the  first  3.5  miles  of  its  surface,  that 
one  half  of  the  remainder  is  contained  in  the  next  3.5  miles, 
and  so  on  until  it  is  so  rare  that  the  kinetic  theory  no  longer 
apphes  without  sensible  modifications. 

33.  The  Escape  of  Atmospheres.  —  Suppose  a  body  is 
projected  upward  from  the  surface  of  the  earth.  The  height 
to  which  it  rises  depends  upon  the  speed  with  which  it  is 
started.  The  greater  the  initial  speed,  the  higher  it  will  rise, 
and  there  is  a  certain  definite  initial  velocity  for  which,  neg- 
lecting the  resistance  of  the  air,  it  will  leave  the  earth  and 
never  return.  This  is  the  velocity  of  escape,  and  for  the 
earth  it  is  a  little  less  than  7  miles  per  second. 

The  molecules  in  the  earth's  atmosphere  may  be  con- 
sidered as  projectiles  which  dart  in  every  direction.     It  has 


70  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  33 

been  seen  that  there  is  a  sm'all  fraction  of  them  which 
move  with  a  velocity  as  great  as  7  miles  per  second.  Half  of 
these  will  move  toward  points  in  the  sky  and  consequently 
would  escape  from  the  earth  if  they  did  not  encounter  other 
molecules.  But  in  view  of  the  great  frequency  of  collisions 
of  molecules,  it  is  evident  that  only  a  very  small  fraction  of 
those  which  move  with  high  velocities  can  escape  from  the 
earth.  However,  it  seems  certain  that  some  molecules  will 
be  lost  in  this  way,  and,  so  far  as  this  factor  is  concerned, 
the  earth's  atmosphere  is  being  continually  depleted.  The 
process  is  much  more  rapid  in  the  case  of  bodies,  such  as 
the  moon,  for  example,  whose  masses  and  attractions  are 
much  smaller,  and  for  which,  therefore,  the  velocity  of 
escape  is  lower. 

It  should  not  be  inferred  from  this  that  the  earth's  at- 
mosphere is  diminishing  in  amount  even  if  possible  replenish- 
ment from  the  rocks  and  its  interior  is  neglected.  When  a 
molecule  escapes  from  the  earth  it  is  still  subject  to  the  attrac- 
tion of  the  sun  and  goes  around  it  in  an  orbit  which  crosses 
that  of  the  earth.  Therefore  the  earth  has  a  chance  of 
acquiring  the  molecule  again  by  collision.  The  only  excep- 
tion to  this  statement  is  when  the  molecule  escapes  with  a 
velocity  so  high  that  the  sun's  attraction  cannot  control  it. 
The  velocity  necessary  in  order  that  the  molecule  shall 
escape  both  the  earth  and  the  sun  depends  upon  its  direction 
of  motion,  but  averages  about  25  miles  per  second  and  cannot 
be  less  than  19  miles  per  second.  But  besides  the  molecules 
that  have  escaped  from  the  earth  there  are  doubtless  many 
others  revolving  around  the  sun  near  the  orbit  of  the  earth. 
These  also  can  be  acquired  by  collision.  The  earth  is  so 
old  and  there  has  been  so  much  time  for  losing  and  acquir- 
ing an  atmosphere,  molecule  by  molecule,  that  probably  an 
equilibrium  has  been  reached  in  which  the  number  of  mole- 
cules lost  equals  the  number  gained.  The  situation  is 
analogous  to  a  large  vessel  of  water  placed  in  a  sealed 
room.     The  water  evaporates  until  the  air  above  it  becomes 


CH.  n,  34]  THE   EARTH  71 

SO  nearly  saturated  that  the  vessel  acquires  as  many  mole- 
cules of  water  vapor  by  collisions  as  it  loses  by  evaporation. 

The  doctrine  of  the  escape  of  atmospheres  imphes  that 
bodies  of  small  mass  will  have  limited  and  perhaps  inappre- 
ciable atmospheres,  and  that  those  of  large  mass  will  have 
extensive  atmospheres.  The  imphcations  of  the  theory  are 
exactly  verified  in  experience.  For  example,  the  moon,  with 
a  mass  ^  that  of  the  earth  and  a  velocity  of  escape  of 
about  1.5  miles  per  sfecond,  has  no  sensible  atmosphere.  On 
the  other  hajid,  Jupiter,  with  a  mass  318  times  that  of  the 
earth  and  a  velocity  of  escape  of  37  miles  per  second,  has 
an  enormous  atmosphere.  These  examples  are  typical  of 
the  facts  furnished  by  all  known  celestial  bodies. 

34.  Effects  of  the  Atmosphere  on  Climate.  —  Aside  from 
the  heat  received  from  the  sun,  the  most  important  factor 
affecting  the  earth's  cUmate  is  its  atmosphere.  It  tends  to 
equaUze  the  temperature  in  three  important  ways,  (a)  It 
makes  the  temperature  at  any  one  place  more  uniform  than 
it  would  otherwise  be,  and  (&)  it  reduces  to  a  large  extent 
the  variations  in  temperature  in  different  latitudes  that 
would  otherwise  exist.  And  (c)  it  distributes  water  over  the 
surface  of  the  earth. 

(a)  Consider  the  day  side  of  the  earth.  The  rays  of  the 
sun  are  partly  absorbed  by  the  atmosphere  and  the  heating 
of  the  earth's  surface  is  thereby  reduced.  The  amount 
absorbed  at  sea  level  is  possibly  as  much  as  40  per  cent. 
Every  one  is  famihar  with  the  fact  that  on  a  mountain, 
above  a  part  of  the  atmosphere,  sunlight  is  more  intense  than 
it  is  at  lower  levels.  But  at  night  the  effects  are  reversed. 
The  heat  that  the  atmosphere  has  absorbed  in  the  daytime 
is  radiated  in  every  direction,  and  hence  some  of  it  strikes 
the  earth  and  warms  it.  Besides  this,  at  night  the  earth 
radiates  the  heat  it  has  received  in  the  daytime.  The  at- 
mosphere above  reflects  some  of  the  radiated  heat  directly 
back  to  the  earth.  Another  portion  of  it  is  absorbed  and 
radiated  in  every  direction,  and  consequently  in  part  back 


72  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  34 

to  the  earth.  In  short,  the  atmosphere  acts  as  a  sort  of 
blanket,  keeping  out  part  of  the  heat  in  the  daytime,  and 
helping  to  retain  at  night  that  which  has  been  received.  Its 
action  is  analogous  to  that  of  a  glass  with  which  the  gardener 
covers  his  hotbed.  The  results  are  that  the  variations  in 
temperature  between  night  and  day  are  reduced,  and  the 
average  temperature  is  raised. 

(6)  The  unequal  heating  of  the  earth's  atmosphere  in 
various  latitudes  is  the  primary  cause  of  the  winds.  The 
warmer  air  moves  toward  the  cooler  regions,  and  the  cold 
air  of  the  higher  latitudes  returns  toward  the  equator.  The 
trade  winds  are  examples  of  these  movements.  Their  im- 
portance will  be  understood  when  it  is  remembered  that 
wind  velocities  of  15  or  20  miles  an  hour  are  not  uncommon, 
and  that  there  is  about  15  pounds  of  air  above  every  square 
inch  of  the  earth's  surface. 

One  of  the  effects  of  the  winds  is  the  production  of  the 
ocean  currents  which  are  often  said  to  be  dominant  factors 
in  modifying  cUmate,  but  which  are,  as  a  matter  of  fact, 
relatively  unimportant  consequences  of  the  air  currents.  A 
south  wind  will  often  in  the  course  of  a  few  hours  raise  the 
temperature  of  the  air  over  thousands  of  square  miles  of 
territory  by  20  degrees,  or  even  more.  In  order  to  raise  the 
temperature  of  the  atmosphere  at  constant  pressure,  over 
one  square  mile  through  20  degrees  by  the  combustion  of  coal 
it  would  be  necessary  to  burn  ten  thousand  tons.  This 
illustration  serves  to  give  some  sort  of  mental  image  of  the 
great  influence  of  air  currents  on  climatic  conditions,  and  if 
it  were  not  for  them,  it  is  probable  that  both  the  equatorial 
and  polar  regions  would  be  uninhabitable  by  man. 

35.  Importance  of  the  Constitution  of  the  Atmosphere.  — 
The  blanketing  effect  of  the  atmosphere  depends  to  a  con- 
siderable extent  on  its  constitution.  Every  one  is  famiUar 
with  the  fact  that  the  early  autumn  frosts  occur  only  when 
the  air  is  clear  and  has  low  humidity.  The  reason  is  that 
water  vapor  is  less  transparent  to  the  earth's  radiations  than 


CH.  II,  36]  THE   EARTH  73 

are  nitrogen  and  oxygen  gas.  On  the  other  hand,  there  is 
not  so  much  difference  in  their  absorption  of  the  rays  that 
come  from  the  sun.  The  reason  is  that  the  very  hot  sun's 
rays  are  largely  of  short  wave  length  (Art.  211) ;  that  is, 
they  are  to  a  considerable  extent  in  the  blue  end  of  the  spec- 
trum, while  the  radiation  from  the  cooler  earth  is  almost 
entirely  composed  of  the  much  longer  heat  rays.  Ordinary 
glass  has  the  same  property,  for  it  transmits  the  sun's  rays 
almost  perfectly,  while  it  is  a  pretty  good  screen  for  the  rays 
emitted  by  a  stove  or  radiator. 

The  water-vapoT  content  of  the  atmosphere  varies  and 
cannot  surpass  a  certain  amount.  But  carbon  dioxide  has 
the  same  absorbing  properties  as  water  vapor,  and  in  spite 
of  the  fact  that  it  makes  up  only  a  very  small  part  of  the 
earth's  atmosphere,  Arrhenius  believes  that  it  has  important 
climatic  effects.  He  concluded  that  if  the  quantity  of  it 
in  the  air  were  doubled  the  climate  would  be  appreciably 
warmer,  and  that  if  half  of  it  were  removed  the  average 
temperature  of  the  earth  would  fall.  Chamberhn  has  shown 
that  there  are  reasons  for  believing  that  the  amount  of 
carbon  dioxide  has  varied  in  long  oscillations,  and  he  sug- 
gested that  this  may  be  the  explanation  of  the  ice  ages,  with 
intervening  warm  epochs,  which  the  middle  latitudes  have 
experienced. 

If  the  effect  of  carbon  dioxide  on  the  cHmate  has  been 
correctly  estimated,  its  production  by  the  recent  enormous 
consumption  of  coal  raises  the  interesting  question  whether 
man  at  last  is  not  in  this  way  seriously  interfering  with  the 
cosmic  processes.  At  the  present  time  about  1,000,000,000 
tons  of  coal  are  mined  and  burned  annually.  In  order  to 
burn  12  pounds  of  coal  32  pounds  of  oxygen  are  required, 
and  the  result  of  the  combustion  is  12  -|-  32  =  44  pounds 
of  carbon  dioxide.  Consequently,  by  the  combustion  of 
coal  there  is  now  annually  produced  by  man  about 
3,670,000,000  tons  of  carbon  dioxide.  On  referring  to  the 
total  amount  of  carbon  dioxide  now  in  the  air  (Art.  28),  it 


74  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ii,  35 

is  seen  that  at  the  present  rate  of  conabustion  of  coal  it  will 
be  doubled  in  800  years.  Consequently,  there  are  grounds 
for  believing  that  modern  industry  may  have  sensible 
climatic  effects  in  a  few  centuries. 

36.  Role  of  the  Atmosphere  in  Life  Processes.  —  Oxygen 
is  an  indispensable  element  in  the  atmosphere  for  all  higher 
forms  of  animal  life.  It  is  taken  into  the  blood  stream 
through  the  lungs  and  is  used  in  the  tissues.  Its  proportion 
in  the  atmosphere  is  probably  not  very  important,  for  it 
seems  probable  that  if  it  had  always  been  much  more  or 
much  less,  animals  would  have  become  adapted  to  the  dif- 
ferent condition.  But  if  the  earth's  crust  had  contained 
enough  material  which  readily  unites  with  oxygen,  such  as 
hydrogen,  silicon,  or  iron,  to  have  exhausted  the  supply,  it 
seems  certain  that  animals  with  warm,  red  blood  could  not 
have  developed.  Such  considerations  are  of  high  impor- 
tance in  speculating  on  the  question  of  the  habitability  of 
other  planets. 

The  higher  forms  of  vegetable  matter  are  largely  composed 
of  carbon  and  water.     The  carbon  is  obtained  from  the  car- 
bon dioxide  in  the  atmosphere.     The  carbon  and  oxygen  are 
2    separated   in    the    cells    of    the 
^    plants,  the  carbon  is  retained,  and 
the  oxygen  is  given  back  to  the  air. 
37.   Refraction  of  Light  by  the 
Atmosphere.  — When  hght  passes 
from  a  rarer  to  a  denser  medium 
it  is  bent  toward  the  perpendic- 
ular to  the  surface  between  the 
two  media,   and  in  general  the 
Fig.  21 .  —  The  refraction  of     greater  the  difference  in  the  densi- 

light. 

ties  of  the  two  media,  the  greater 
is  the  bending,  which  is  called  refraction.  Thus,  in  Fig.  21, 
the  ray  I  which  strikes  the  surface  of  the  denser  medium 
at  A  is  bent  from  the  direction  AB  toward  the  perpendicular 
to  the  surface  AD  and  takes  the  direction  AG. 


CH.  II,  37] 


THE   EARTH 


75 


Now  consider  a  ray  of  light  striking  the  earth's  atmosphere 
obliquely.  The  density  of  the  air  increases  from  its  outer 
borders   to    the    surface 

of    the    earth.      Conse-  '  ■^' 

quently,  a  ray  of  Ught  is 
bent  more  and  more  as  it 
proceeds  down  through 
the  air.  Let  I,  Fig.  22, 
represent  a  ray  of  hght 
coming  from  a  star  S  to 
an  observer  at  0.  The 
star  is  really  in  the  direc- 
tion OS",  but  it  appears  to  be  in  the  direction  OS'  from 
which  the  light  comes  when  it  strikes  the  observer's  eye.  The 
angle  between  OS"  and  OS'  is  the  angle  of  refraction.  It  is 
zero  for  a  star  at  the  zenith  and  increases  to  a  httle  over 
one-half  of  a  degree  for  one  at  the  horizon.     For  this  reason  a 


Fig.  22.  —  Refraction    of   light   by  the 
earth's  atmosphere. 


f IG.  23.  —  The  sun  is  apparently  flattened  by  refraction  when  it  is  on  the 

horizon. 

celestial  body  apparently  rises  before  it  is  actually  above  the 
horizon,  and  is  visible  until  after  it  has  really  set.  If  the 
sun  or  moon  is  on  the  horizon,  its  bottom  part  is  apparently 
raised  more  than  its  top  part  by  refraction,  so  that  it  seems 
to  be  flattened  in  the  vertical  direction,  as  is  shown  in  Fig.  23. 


76  AN   IlsrTRODUCTION   TO   ASTRONOMY     [ch.  ii,  38 

38.  The  Twinkling  of  the  Stars.  —  The  atmosphere  is  not 
only  of  variable  density  from  its  highest  regions  to  the  sur- 
face of  the  earth,  but  it  is  always  disturbed  by  waves  which 
cause  the  density  at  a  given  point  to  vary  continually. 
These  variations  in  density  cause  constant  small  changes  in 
the  refraction  of  light,  and  consequently  alterations  in  the 
direction  from  which  the  light  appears  to  come.  When 
the  source  is  a  poirft  of  hght,  as  a  star,  it  twinkles  or  scintil- 
lates. The  twinkling  of  the  stars  is  particularly  noticeable 
in  winter  time  on  nights  when  the  air  is  cold  and  unsteady. 
The  variation  in  refraction  is  dffierent  for  different  colors, 
and  consequently  when  a  star  twinkles  it  flashes  sometimes 
blue  or  green  and  at  other  times  red  or  yellow.  Objects  that 
have  disks,  even  though  they  are  too  small  to  be  discerned 
with  the  unaided  eye,  appear  much  steadier  than  stars  because 
the  irregular  refractions  from  various  parts  seldom  agree  in 
direction,  and  consequently  do  not  displace  the  whole  object. 

IV.  QUESTIONS 

1.  What  is  the  weight  of  the  air  in  a  room  16  feet  square  and 
10  feet  high? 

2.  How  many  pounds  of  air  pass  per  minute  through  a  windmill 
12  feet  in  diameter  in  a  breeze  of  20  miles  per  hour  ? 

3.  Compute  the  approximate  total  atmospheric  pressure  to  which 
a  person  is  subject. 

4.  What  is  the  density  of  the  air,  compared  to  its  density  at  the 
surface,  at  heights  of  50,  100,  and  500  miles,  the  density  being  deter- 
mined by  the  law  given  at  the  end  of  Art.  32  ?  This  gives  an  idea 
of  the  density  required  for  the  phenomena  of  twilight,  of  meteors, 
and  of  aurorsB. 

5.  Draw  a  diagram  showing  the  earth  and  its  atmosphere  to  scale. 

6.  The  earth's  mass  is  slowly  growing  by  the  acquisition  of 
meteors ;  if  there  is  nothing  to  offset  this  growth,  will  its  atmosphere 
have  a  tendency  to  increase  or  to  decrease  in  amount  ? 

7.  If  the  earth's  atmosphere  increases  or  decreases,  as  the  case 
may  be,  what  will  be  the  effect  on  the  mean  temperature,  the  daily 
range  at  any  place,  and  the  range  over  the  earth's  whole  surface  ? 

8.  If  the  earth's  surface  were  devoid  of  water,  what  would  be  the 
effect  on  the  mean  temperature,  the  daily  range  at  any  place,  and 
the  range  over  its  whole  surface  ? 


CHAPTER  III 
THE   MOTIONS   OF   THE   EARTH 


I.   The  Rotation  of  the  Eakth 

39.  The  Relative  Rotation  of  the  Earth. —  The  most 
casual  observer  of  the  heavens  has  noticed  that  not  only 
the  sun  and  moon,  but  also  the  stars,  rise  in  the  east,  pass 
across  the  sky,  and  set  in 
the  west.  At  least  this  is 
true  of  those  stars  which 
cross  the  meridian  south 
of  the  zenith.  Figure  24 
is  a  photograph  of  Orion 
in  which  the  telescope  was 
kept  fixed  while  the  stars 
passed  in  front  of  it,  and 
the  horizontal  streaks  are 
the  images  traced  out  by 
the  stars  on  the  photo- 
graphic plate. 

The  stars  in  the  north- 
ern heavens  describe  circles 
around  the  north  pole  of 
the, sky  as  a  center.  Two  hours  of  observation  of  the  posi- 
tion of  the  Big  Dipper  will  show  the  character  of  the  motion 
very  clearly.  Figure  25  shows  circumpolar  star  trails  secured 
by  pointing  a  fixed  telescope  toward  the  pole  star  and  giving 
an  exposure  of  a  little  over  an  hour.  The  conspicuous 
streak  a  little  below  and  to  the  left  of  the,  center  is  the 
trail  of  the  pole  star,  which  therefore  is  not  exactly  at  the 
pole  of  the  heavens.      A  comparison  of  this  picture  with 

77 


Fig.  24.  — •  Star  trails  of  brighter  stars 
in  Orion  (Barnard). 


78 


AN   INTRODUCTION   TO  ASTRONOMY     [ch.  hi.  39 


the  northern  sky  will  show  that  most  of  the  stars  whose 
trails  are  seen  are  quite  invisible  to  the  unaided  eye. 

Since  all  the  heavenly  bodies  rise  in  the  east  (except  those 
so  near  the  pole  that  they  simply  go  around  it),  travel  across 

the  sky,  and  set 
in  the  west,  to 
reappear  again  in 
the  east,  it  fol- 
lows that  either 
they  go  around 
the  earth  from 
east  to  west,  or 
the  earth  turns 
from  west  to 
east.  So  far  as 
the  simple  mo- 
tions of  the  sun, 
moon,  and  stars 
are  concerned 
both  hypotheses 
are  in  perfect 
harmony  with 
the  observations, 
and  it  is  not  pos- 
sible to  decide 
which  of  them  is  correct  without  additional  data.  All  the 
apparent  motions  prove  is  that  there  is  a  relative  motion 
of  the  earth  with  respect  to  the  heavenly  bodies. 

It  is  often  supposed  that  the  ancients  were  unscientific, 
if  not  stupid,  because  they  believed  that  the  earth  was  fixed 
and  that  the  sky  went  around  it,  but  it  has  been  seen  that 
so  far  as  their  data  bore  on  the  question  one  theory  was  as 
good  as  the  other.  In  fact,  not  all  of  them  thought  that 
the  earth  was  fixed.  The  earliest  philosopher  who  is  known 
to  have  believed  in  the  rotation  of  the  earth  was  Philolaus, 
a  Pythagorean,  who  lived  in  the  fifth  century  b.c.     His 


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iiii 

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Bw^' 

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Fig.  25.  —  Circumpolar  star  trails  (Ritchey) . 


CH.  Ill,  40]      THE   MOTIONS   OF   THE   EARTH  79 

ideas  were  more  or  less  mystical,  but  they  seem  to  have  had 
some  influence,  for  they  were  quoted  by  Copernicus  (1473- 
1543)  in  his  great  work  on  the  theory  of  the  motions  in  the 
solar  system.  Aristotle  (384-322  B.C.)  recognized  the  fact 
that  the  apparent  motions  of  the  stars  can  be  explained 
either  by  their  revolution  around  the  earth,  or  by  the  rota- 
tion of  the  earth  on  its  axis.  Aristarchus  of  Samos  (310- 
250  B.C.)  made  the  clearest  statements  regarding  both  the 
rotation  and  the  revolution  of  the  earth  of  any  philosopher 
of  antiquity.  But  Hipparchus  (180-110  B.C.),  who  was  the 
greatest  astronomer  of  antiquity,  and  whose  discoveries 
were  very  numerous  and  valuable,  beheved  in  the  fixity  of 
the  earth.  He  was  followed  in  this  opinion  by  Ptolemy 
(100-170  A.D.)  and  every  other  astronomer  of  note  down  to 
Copernicus,  who  believed  the  earth  rotated  and  revolved 
around  the  sun. 

40.  The  Laws  of  Motion.  —  One  method  of  attacking 
the  question  of  whether  or  not  any  particular  body,  such  as 
the  earth,  moves  is  to  consider  the  laws  of  motion  of  bodies 
in  general,  and  then  to  answer  it  on  the  basis  of,  and  in 
harmony  with,  these  laws.  The  laws  of  nature  are  in  a' 
fundamental  respect  different  from  civil  laws,  and  it  is  un- 
fortunate that  the  same  term  is  used  for  both  of  them.  A 
civil  law  prescribes  or  forbids  a  mode  of  conduct,  with  pen- 
alties if  it  is  violated.  It  can  be  violated  at  pleasure  if  one 
is  willing  to  run  the  chance  of  suffering  the  penalty.  On 
the  other  hand,  a  law  of  nature  does  not  prescribe  or  compel 
anything,  but  is  a  description' of  the  way  all  phenomena  of 
a  certain  class  succeed  one  another. 

The  laws  of  motion  are  statements  of  the  way  bodies 
actually  move.  They  were  first  given  by  Newton  in  1686, 
altholigh  they  were  to  some  extent  understood  by  his  prede- 
cessor Gahleo.  Newton  called  them  axioms  although  they 
are  by  no  means  self-evident,  as  is  proved  by  the  fact  that 
for  thousands  of  years  they  were  quite  unknown.  The  laws, 
essentially  as  Newton  gave  them,  are : 


80        AN   INTRODUCTION   TO  ASTRONOMY     [ch.  hi,  40 

Law  I.  Every  body  continues  in  its  state  of  rest,  or  of  uni- 
form motion  in  a  straight  line,  unless  it  is  compelled  to  change 
that  state  by  an  exterior  force  acting  upon  it. 

Law  II.  The  rate  of  change  of  motion  of  a  body  is  directly 
proportional  to  the  force  applied  to  it  and  inversely  propor- 
tional to  its  mass,  and  the  change  of  motion  takes  place  in  the 
direction  of  the  line  in  which  the  force  acts. 

Law  III.  To  every  action  there  is  an  equal  and  oppositely 
directed  reaction ;  or,  the  mutual  actions  of  two  bodies  are  al- 
ways equal  and  oppositely  directed. 

The  importance  of  the  laws  of  motion  can  be  seen  from  the 
fact  that  every  astronomical  and  terrestrial  phenomenon 
involving  the  motion  of  matter  is  interpreted  by  using  them 
as  a  basis.  They  are,  for  example,  the  foundation  of  all 
mechanics.  A  little  reflection  will  lead  to  the  conclusion 
that  there  are  few,  if  indeed  any,  phenomena  that  do  not  in 
some  way,  directly  or  indirectly,  depend  upon  the  motion 
of  matter. 

The  first  law  states  the  important  fact  that  if  a  body  is  at 
rest  it  will  never  begin  to  move  unless  some  force  acts  upon 
it,  and  that  if  it  is  in  motion  it  will  forever  move  with  uniform 
speed  in  a  straight  Une  unless  some  exterior  force  acts  upon 
it.  In  two  respects  this  law  is  contradictory  to  the  ideas 
generally  maintained  before  the  time  of  Newton.  In  the 
first  place,  it  had  been  supposed  that  bodies  near  the  earth's 
surface  would  descend,  because  it  was  natural  for  them  to  do 
so,  even  though  no  forces  were  acting  upon  them.  In  the 
second  place,  it  had  been  supposed  that  a  moving  body  would 
stop  unless  some  force  were  continually  applied  to  keep  it 
going.  These  errors  kept  the  predecessors  of  Newton  from 
getting  any  satisfactory  theories  regarding  the  motions  of 
the  heavenly  bodies. 

The  second  law  defines  how  the  change  of  motion  of  a 
body,  in  both  direction  and  amount,  depends  upon  the  ap- 
plied force.  It  asserts  what  happens  when  any  force  is  act- 
ing, and  this  means  that  the  statement  is  true  whether  or 


CH.  Ill,  40]      THE   MOTIONS   OF   THE   EARTH  81 

not  there  are  other  forces.     In  other  words,  th6  momentary 

effects  of  forces  can  be  considered  independently  of  one 

another.     For  example,  if  two  forces,  PA  and  PB  in  Fig. 

26,  are  acting  on  a  body  at  P,  it  will  move  in  the  direction 

PA  just  as  though  PB 

were  not  acting  on  it,  /  ^^^-'^^ 

and  it  will  move  in  the  /  ^        /' 

direction    PB    just    as  /      ^^^--^'''''^ 

though    PA    were    not        /^--^^^ 

acting  on  it.   The  result    p  ^'■^ 

is    that    when    they    are        Fig.  26.  —  The  parallelogram  of  forces. 

both  acting  it  will  go  from  P  to  C  along  PC.  Since  PACB 
is  a  parallelogram,  this  is  called  the  parallelogram  law  of 
the  composition  of  forces. 

The  first  two  laws  refer  to  the  motion  of  a  single  body; 
the  third  expresses  the  way  in  which  two  bodies  act  on  each 
other.  It  means  essentially  that  if  one  body  changes  the 
state  of  motion  of  another  body,  its  own  state  of  motion  is 
also  changed  reciprocally  in  a  definite  way.  The  term 
"  action  "  in  the  law  means  the  mass  times  the  rate  of  change 
of  motion  (acceleration)  of  the  body.  Hence  the  third  law 
might  read  that  if  two  bodies  act  on  each  other,  then  the 
product  of  the  mass  and  acceleration  in  one  is  equal  and 
opposite  to  the  product  of  the  mass  and  acceleration  in  the 
other.  This  is  a  complete  statement  of  the  way  two  bodies 
act  upon  each  other.  But  the  second  law  states  that  the 
product  of  the  mass  and  acceleration  of  a  body  is  propor- 
tional to  the  force  acting  on  it.  Hence  it  follows  that  the 
third  law  might  read  that  if  two  bodies  act  on  each  other, 
then  the  force  exerted  by  the  first  on  the  second  is  equal 
and  opposite  to  the  force  exerted  by  the  second  on  the  first. 
This  statement  is  not  obviously  true  because  it  seems  to 
contradict  ordinary  experience.  For  example,  the  law  states 
that  if  a  strong  man  and  a  weak  man  are  pulling  on  a  rope 
(weight  of  the  rope  being  neglected)  against  each  other,  the 
strong  man  cannot  pull  any  more  than  the  weak  man.     The 


82         AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  40 

reason  is,  of  course,  that  the  weak  man  does  not  give  the 
strong  one  an  opportunity  to  use  his  full  strength.  If  the 
strong  man  is  heavier  than  the  weak  one  and  pulls  enough, 
he  will  move  the  latter  while  he  remains  in  his  tracks.  This 
seems  to  contradict  the  statement  of  the  law  in  terms  of 
the  acceleration;  but  the  contradiction  disappears  when  it 
is  remembered  that  the  men  are  subject  not  only  to  the  forces 
they  exert  on  each  other,  but  also  to  their  friction  with  the 
earth.  If  they  were  in  canoes  in  open  water,  they  would 
both  move,  and,  if  the  weights  of  the  canoes  were  included, 
their  motions  would  be  in  harmony  with  the  third  law. 

Since  the  laws  of  motion  are  to  be  used  fundamentally  in 
considering  the  motion  of  the  earth,  the  question  of  their 
truth  at  once  arises.  When  they  are  applied  to  the  motions 
of  the  heavenly  bodies,  everything  becomes  orderly.  Be- 
sides this,  they  have  been  illustrated  millions  of  times  in 
ordinary  experience  on  the  earth  and  they  have  been  tested 
in  laboratories,  but  nothing  has  been  found  to  indicate  they 
are  not  in  harmony  with  the  actual  motions  of  material  bodies. 
In  fact,  they  are  now  supported  by  such  an  enormous  mass 
of  experience  that  they  are  among  the  most  trustworthy  con- 
clusions men  have  reached. 

41.  Rotation  of  the  Earth' Proved  by  Its  Shape. —  The 
shape  of  the  earth  can  be  determined  without  knowing  whether 
or  not  it  rotates.  The  simple  measurements  of  arcs  (Art.  12) 
prove  that  the  earth  is  oblate. 

It  can  be  shown  that  it  follows  from  the  laws  of  motion 
and  the  law  of  gravitation  that  the  earth  would  be  spherical 
if  it  were  not  rotating.  Since  it  is  not  spherical,  it  must  be 
rotating.  Moreover,  it  follows  from  the  laws  of  motion 
that  if  it  is  rotating  it  will  be  bulged  at  the  equator.  Hence 
the  oblateness  of  the  earth  proves  that  it  rotates  and  deter- 
mines the  position  of  its  axis,  but  does  not  determine  in 
which  direction  it  turns. 

42.  Rotation  of  the  Earth  Proved  by  the  Eastward  Devi- 
ation of  Falling  Bodies.  —  Let  OP,  Fig.   27,   represent  a 


CH.  Ill,  42]      THE   MOTIONS   OP   THE   EARTH 


83 


Fig.  27.  —  The  eastward  deviation  of  falling  bodies 
proves  the  eastward  rotation  of  the  earth. 


tower  from  whose  top  a  ball  is  dropped.  Suppose  that  while 
the  ball  is  falling  to  the  foot  of  the  tower  the  earth  rotates 
through  the  angle  QEQ'.  The  top  of  the  tower  is  carried 
from  P  to  P',  and  its  foot  from  0  to  0'.  The  distance  PP' 
is  somewhat  greater  than  the  distance  00'.  Now  consider 
the  falling  body. 
It  tends  to  move 
in  the  direction 
PP'  in  accord- 
ance with  the  first 
law  of  motion  be- 
cause, at  the  time 
it  is  dropped,  it 
is  carried  in  this 
direction  by  the 
rotation  of  the 
earth.  Moreover, 
PP'  is  the  dis- 
tance through  which  it  would  be  carried  if  it  were  not 
dropped.  But  the  earth's  attraction  causes  it  to  descend, 
and  the  force  acts  at  right  angles  to  the  Une  PP'.  There- 
fore, by  the  second  law  of  motion,  the  attraction  of  the  earth 
does  not  have  any  influence  on  the  motion  in  the  direction 
PP'.  Consequently,  while  it  is  descending  it  moves  in  a 
horizontal  direction  a  distance  equal  to  PP'  and  strikes 
the  surface  at  0"  to  the  east  of  the  foot  of  the  tower  0'. 
The  eastward  deviation  is  the  distance  O'O".  The  small 
diagram  at  the  right  shows  the  tower  and  the  path  of  the 
falling  body  on  a  larger  scale. 

The  foregoing  reasoning  has  been  made  on  the  assumption 
that  the  earth  rotates  to  the  eastward.  The  question  arises 
whether  the  conclusions  are  in  harmony  with  experience. 
The  experiment  for  determining  the  deviation  of  falling  bodies 
is  complicated  by  air  currents  and  the  resistance  of  the  air. 
Furthermore,  the  eastward  deviation  is  very  small,  being 
only  1.2  inches  for  a  drop  of  500  feet  in  latitude  40°.    In 


84         AN   INTRODUCTION   TO  ASTRONOMY     [ch.  hi,  42 


spite  of  these  difficulties,  the  experiment  for  moderate  heights 
proves  that  the  earth  rotates  to  the  eastward.  Father  Hagen, 
of  Rome,  has  devised  an  apparatus,  having  analogies  with 
Atwood's  machine  in  physics,  which  avoids  most  of  the  dis- 
turbances to  which  a  freely  falling  body  is  subject.  The 
largest  free  fall  so  far  tried  was  in  a  vertical  mine  shaft,  near 
Houghton,  Mich.,  more  than  4000  feet  deep.  In  spite 
of  the  fact  that  the  diameter  of  the  mine  shaft  was  many 
times  the  deviation  for  that  distance,  the  experiment  utterly 
failed  because  the  balls  which  were  dropped  never  reached 
the  bottom.  It  is  probable  that  when  they  had  fallen  far 
enough  to  acquire  high  speed  the  air  packed  up  in  front  of 
them  until  they  were  suddenly  deflected  far  enough  from 
their  course  to  hit  the  walls  and  become  imbedded. 

43.   Rotation  of  the  Earth  Proved  by  Foucault's  Pendulum. 
—  One  of  the  most  ingenious  and  convincing  experiments 

for  proving  the 
rotation  of  the 
earth  was  devised 
in  1851  by  the 
French  physicist 
Foucault.  It  de- 
pends upon  the 
fact  that  accord- 
ing to  the  laws  of 
motion  a  freely 
swinging  pendu- 
lum tends  con- 
stantly to  move  in 
the  same  plane." 
Suppose  a  pendulum  suspended  at  0,  Fig.  28,  is  started 
swinging  in  the  meridian  OQ.  Let  OF  be  the  tangent  at  0 
drawn  in  the  plane  of  the  meridian.  After  a  certain  interval 
the  meridian  OQ  will  have  rotated  to  the  position  O'Q'. 
The  hne  O'V  is  drawn  parallel  to  the  line  OV.  Conse- 
quently the  pendulum  will  be  swinging  in  the  plane  EO'V. 


Fig.  28.  —  The  Foucault  pendulum. 


CH.  Ill,  44]      THE   MOTIONS   OF  THE   EARTH 


85 


The  tangent  to  the  meridian  at  0'  is  O'F.  Consequently, 
the  angle  between  this  line  and  the  plane  in  which  the 
pendulum  will  be  swinging  is  V'O'V,  which  equals  OVO'. 
That  is,  the  angle  at  V  between  the  meridian  tangents  equals 
the  apparent  deviation  of  the  plane  of  the  pendulum  from  the 
meridian.  For  points  in  the  northern  hemisphere  the  devi- 
ation is  from  a  north-and-south  direction  toward  a  northeast- 
and-southwest  direction.  The  angle  around  the  cone  at  V 
equals  the  total  deviation  in  one  rotation  of  the  earth.  If 
0  is  at  the  earth's  pole,  the  daily  deviation  is  360  degrees. 
If  0  is  on  the  ea,rth's  equator,  the  point  V  is  infinitely  far 
away  and  the  deviation  is  zero. 

Foucault  suspended  a  heavy  iron  ball  by  a  steel  wire  about 
200  feet  long,  and  the  deviation  became  evident  in  a  few 
minutes.  The  experiment  is  very  simple  and  has  been  re- 
peated in  many  pla)ces.  It  proves  that  the  earth  rotates 
eastward,  and  the  rate  of  deviation  of  the  pendulum  proves 
that  the  relative  motion  of  the  earth  with  respect  to  the 
stars  is  due  entirely  to  its  rotation  and  not  at  all  to  the 
motions  of  the  stars  around  it. 

44.  Consequences  of  the  Earth's  Rotation.  —  An  itripor- 
tant  consequence  of  the  earth's  rotation  is  the  direction  of 
air  currents  at 
considerable  dis- 
tances from  the 
equator  in  both 
northern  and 
southern  lati- 
tudes. Suppose 
the  unequal  heat- 
ing of  the  atmos- 
phere causes  a 
certain  portion  of 
it  to  move  north- 
ward from  0,  Fig.  29,  with  such  a  velocity  that  if  the 
earth  were  not  rotating,  it  would  arrive  at  A  in  a  certain 


— 

/y. 

X7 

/!■ 

^  1 

/ 

/  ^ 

7 

'      / 

-^ 

t' 

JSi- 



Fig.  29.  —  The  deviation  of  air  currents. 


86         AN   INTRODUCTION   TO  ASTRONOMY     [ch.  hi,  44 

interval  of  time.  Suppose  that  in  this  interval  of  time  the 
meridian  OQ  rotates  to  the  position  O'Q'.  Hence  the  mass 
of  air  under  consideration  actually  had  the  velocities  OA  and 
00'  when  it  started  from  0,  the  former  with  respect  to  the 
surface  of  the  earth  and  the  latter  because  of  the  rotation  of 
the  earth.  By  the  laws  of  motion  these  motions,  being  at 
right  angles  to  each  other,  are  mutually  independent,  and 
the  air  will  move  over  both  distances  during  the  interval  of 
time  and  arrive  at  the  point  A",  which  is  east  of  A'.  Con- 
sequently, the  mass  of  air  that  started  straight  northward 
with  respect  to  the  surface  of  the  earth  along  the  meridian 
OA  will  have  deviated  eastward  by  the  amount  A'A". 

The  deviation  for  northward  motion  in  the  northern 
hemisphere  is  toward  *the  east ;  for  southward  motion,  it 
is  toward  the  west.  In  both  cases  it  is  toward  the  right. 
For  similar  reasons,  in  the  southern  hemisphere  the  devia- 
tion is  toward  the  left. 

The  deviations  in  the  directions  of  air  currents  are  evi- 
dently greater  the  higher  the  latitude,  because  near  the  poles 
a  given  distance  along  the  earth's  surface  corresponds  to 
an  almost  equal  change  in  the  distance  from  the  axis  of 
rotation,  while  at  the  equator  there  is  no  change  in  the  dis- 
tance from  the  earth's  axis.  It  might  be  supposed  that  in 
middle  latitudes  a  moderate  northward  or  southward  dis- 
placement of  the  air  would  cause  no  appreciable  change  in 
its  direction  of  motion.  But  a  point  on  the  equator  moves 
eastward  at  the  rate  of  over  1000  miles  an  hour,  at  latitude 
60  degrees  the  eastward  velocity  is  half  as  great,  and  at  the 
pole  it  is  zero.  If  it  were  not  for  friction  with  the  earth's 
surface,  a  mass  of  air  moving  from  latitude  40  degrees  to 
latitude  45  degrees,  a  distance  less  than  350  miles,  would 
acquire  an  eastward  velocity  with  respect  to  the  surface  of 
the  earth  of  over  40  miles  an  hour.  The  prevailing  winds 
of  the  northern  hemisphere  in  middle  latitudes  are  to  the 
northeast,  and  the  eastward  component  has  been  found  to 
be  strong  for  the  very  high  currents. 


CH.  Ill,  45]      THE   MOTIONS  OF  THE   EARTH  87 

Obviously  the  same  principles  apply  to  water  currents 
and  to  air  currents.  Consequently  water  currents,  such  as 
rivers,  tend  to  deviate  toward  the  right  in  the  northern 
hemisphere.  It  has  been  found  by  examining  the  Missis- 
sippi and  Yukon  rivers  that  the  former  to  some  extent, 
and  the  latter  to  a  much  greater  extent,  on  the  whole  scour 
their  right-hand  banks. 

All  the  proofs  of  the  earth's  rotation  so  far  given  depend 
upon  the  laws  of  motion.  There  is  one  independent  reason 
for  believing  the  earth  rotates,  though  it  falls  a  little  short 
of  proof.  It  has  been  found  by  observations  involving 
only  geometrical  principles  that  the  sun,  moon,  and  planets 
are  comparable  to  the  earth  in  size,  some  being  larger  and 
others  smaller.  Direct  observations  with  the  telescope  show 
'  that  a  number  of  these  bodies  rotate  on  their  axes,  the  re- 
mainder being  either  very  remote  or  otherwise  unfavorably 
situated  for  observation.  The  conclusion  by  analogy  is 
that  the  -earth  also  rotates. 

45.  The  Uniformity  of  the  Earth's  Rotation.  —  It  follows 
from  the  laws  of  motion,  and  in  particular  from  the  first 
law,  that  if  the  earth  were  subject  to  no  external  forces  and 
were  invariable  in  size,  shape,  and  distribution  of  mass,  it 
would  rotate  on  its.  axis  with  absolute  uniformity.  Since 
the  earth  is  a  fundamental  means  of  measuring  time  its 
rotation  cannot  be  tested  by  clocks.  Its  rotation  might  be 
compared  with  other  celestial  phenomena,  but  then  the 
question  of  their  uniformity  would  arise.  The  only  re- 
course is  to  make  an  examination  of  the  possible  forces  and 
changes  in  the  earth  which  are  capable  of  altering  the  rate 
of  its  rotation. 

The  earth  is  subject  to  the  attractions  of  the  sun,  moon, 
and  planets.  But  these  attractions  do  not  change  its  rate 
of  rotation  because  the  forces  pulling  on  opposite  sides 
balance,  just  as  the  earth's  attraction  for  a  rotating  wh^el 
whose- plane  is  vertical  neither  retards  nor  accelerates  its 
motion. 


88         AN   INTRODUCTION    TO   ASTRONOMY     [ch.  hi,  45 

The  earth  is  struck  by  milhons  of  small  meteors  daily 
coming  in  from  all  sides.  They  virtually  act  as  a  resisting 
medium  and  sHghtly  retard  its  rotation,  just  as  a  top  spin- 
ning in  the  air  is  retarded  by  the  molecules  impinging  on  it. 
But  'the  mass  of  the  earth  is  so  large  and  the  meteors  are  so 
small  that,  at  their  present  rate  of  infall,  the  length  of  the 
day  cannot  be  changed  by  this  cause  so  much  as  a  second  in 
100,000,000  years. 

The  moon  and  the  sun  generate  tides  in  the  water  around 
the  earth  and  the  waves  beat  in  upon  the  shores  and  are 
gradually  destroyed  by  friction.  The  energy  of  the  waves 
is  transformed  into  heat.  This  means  that  something  else 
has  lost  energy,  and  a  mathematical  treatment  of  the  sub- 
ject shows  that  the  earth  has  suffered  the  loss.  Conse- 
quently its  rotation  is  diminished.  But  as  great  and  irre- 
sistible as  the  tides  may  be,  their  energies  are  insignificant 
compared  to  that  of  the  rotating  earth,  and  according  to  the 
work  of  MacMillan  the  day  is  not  increasing  in  length  from 
this  cause  more  than  one  second  in  500,000  years. 

Before  discussing  the  effects  of  a  change  in  the  size  of  the 
earth  or  in  the  distribution  of  its  mass,  it  is  necessary  to 
explain  a  very  important  property  of  the  motion  of  rotating 
bodies.  It  can  be  shown  from  the  laws  of  motion  that  if 
a  body  is  not  subject  to  any  exterior  forces,  its  total  quantity 
of  rotation  always  remains  the  same  no  matter  what  changes 
may  take  place  in  the  body  itself.  The  quantity  of  rotation 
of  a  body,  or  moment  of  momentum,  as  it  is  technically  called 
in  mechanics,  is  the  sum  of  the  rotations  of  all  its  parts. 
The  rotation  of  a  single  part,  or  particle,  is  the  product  of 
its  mass,  its  distance  from  the  axis  of  rotation  passing 
through  the  center  of  gravity  of  the  body,  and  the  speed 
with  which  it  is  moving  at  right  angles  to  the  Une  joining  it 
to  the  axis  of  rotation.  It  can  be  shown  that  in  the  case 
of  a  body  rotating  as  a  solid,  the  quantity  of  rotation  is 
proportional  to  the  product  of  the  square  of  the  radius  and 
the  angular  velocity  of  rotation,  the  angular  velocity  of 


CH.  Ill,  46]      THE   MOTIONS   OF   THE   EARTH  89 

rotation  being  the  angle  through  which  the  body  turns  in 
a  unit  of  time. 

Now  apply  this  principle  of  the  conservation  of  the  mo- 
ment of  momentum  to  the  earth.  If  it  should  lose  heat  and 
shrink  so  that  its  radius  were  diminished  in  length,  then  the 
angular  velocity  of  rotation  would  increase,,  for  the  product 
of  the  square  of  the  radius  and  the  rate  of  rotation  must 
be  constant.  On  the  other  hand,  if  the  radio-active  sub- 
stances in  the  earth  should  cause  its  temperature  to  rise  and 
its  radius  to  expand,  then  the  rate  of  rotation  would  de- 
crease. Neither  of  these  causes  can  make  a  sensible  change 
in  the  rotation  in  1,000,000  years.  Similarly,  if  a  river 
rising  in  low  latitudes  should  carry  sediment  to  higher  lati- 
tudes and  deposit  it  nearer  the  earth's  axis,  then  the  rate 
of  rotation  of'  the  earth  would  be  increased.  While  such 
factors  are  theoretically  effective  in  producing  changes  in 
the  rotation  of  the  earth,  from  a-  practical  point  of  view 
they  are  altogether  negligible. 

It  follows  from  this  discussion  that  there  are  some  influ- 
ences tending  to  decrease  the  rate  of  the  earth's  rotation, 
and  others  tending  to  increase  it,  but  that  they  are  all  so 
sinall  as  to  have  altogether  inappreciable  effects  even  in  a 
period  as  long  as  100,000  years. 

46.  The  Variation  of  Latitude.  —  It  was  mentioned  in 
connection  with  the  discussion  of  the  rigidity  of  the  earth 
(Arts.  25,  26),  that  its  axis  of  rotation  is  not  exactly  fixed. 
This  does  not  mean  that  the  direction  of  the  axis  changes, 
but  that  the  position  of  the  earth  itself  changes  so  that  its 
axis  of  rotation  continually  pierces  different  parts  of  its 
surface.  That  is,  the  poles  of  the  earth  are  not  fixed  points 
on  its  surface.  Since  the  earth's  equator  is  90  degrees  from 
its  poles,  the  position  of  the  equator  also  continually  changes. 
Therefore  the  latitude  of  any  fixed  point  on  the  surface  of 
the  earth  undergoes  continual  variation.  The  fact  was 
discovered  by  very  accurate  determinations  of  latitude,  and 
for  this  reason  is  known  as  the  variation  of  latitude. 


90         AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  46 

The  pole  wanders  from  its  mean  position  not  more  than 
30  feet,  corresponding  to  a  change  of  latitude  of  0.3  of  a 
second  of  arc.  This  is  such  a  small  quantity  that  it  can  be 
measured  only  by  the  most  refined  means,  and  accounts 


,  Fig.   30.  —  The  position  of  the  pole  from  1906  to  1913. 

for  the  failure  to  discover  it  until  the  work  of  Chandler  and 
Kiistner  about  1885. 

In  1891  Chandler  took  up  the  problem  of  finding  from 
the  observations  how  the  pole  actually  moves.  The  varia- 
tion in  its  position  is  very  complicated,  Fig.  30  showing  it 


CH.  Ill,  46]      THE   MOTIONS  OF  THE   EARTH  91 

from  1906  to  1913.  Chandler  found  that  this  compHcated 
motion  is  the  result  of  two  simpler  ones.  The  first  is  a 
yeady  motion  in  an  ellipse  (Art.  53)  whose  longest  radius  is 
14  feet  and  shortest  radius  4  feet;  and  the  second  is  a 
motion  in  a  circle  of  radius  15  feet,  which  is  described  in 
about  428  days.  More  recent  discussions,  based  on  observa- 
tions secured  by  the  cooperation  of  the  astronomers  of  several 
countries,  have  modified  these  results  to  some  extent  and 
have  added  other  minor  terms. 

The  problem  is  to  account  for  the  variation  of  latitude 
and  for  the  different  periods.  Unless  a  freely  rotating  ob- 
late rigid  body  is  started  turning  exactly  around  its  shortest 
axis,  it  will  undergo  an  oscillation  with  respect  to  its  axis 
of  rotation  in  a  period  which  depends  upon  its  figure,  mass, 
and  speed  of  rotation.  Hence  it  might  be  supposed  that 
the  earth  in  some  way  originally  started  rotating  in  this 
manner.  But  since  the  earth  is  not  perfectly  rigid  and  un- 
yielding, friction  would  in  the  course  of  time  destroy  the 
wabbling.  In  view  of  the  fact  that  the  earth  is  certainly 
many  millions  of  years  old,  it  seems  that  friction  should 
long  ago  have  reduced  its  rotation  to  sensible  uniformity 
around  a  fixed  axis,  and  this  is  true  unless  it  is  very  elastic 
instead  of  being  somewhat  viscous.  The  tide  experiment 
(Art.  25)  proves  that  the  earth  is  very  elastic  and  suggests 
that  perhaps  the  earth's  present  irregularities  of  rotation 
have  been  inherited  from  greater  ones  produced  at  the  time 
of  its  origin,  possibly  by  the  falling  together  of  scattered 
meteoric  masses.  But  the  fact  that  the  earth  has  two  dif- 
ferent variations  of  latitude  of  almost  equal  magnitude  is 
opposed  to  this  conclusion.  The  one  which  has  the  period 
of  a  year  is  probably  produced  by  meteorological  causes,  as 
Jeffreys  infers  from  a  quantitative  examination  of  the  ques- 
tion. The  one  whose  period  is  428  days,  the  natural  period 
of  variation  of  latitude  for  a  body  having  the  dynamical 
properties  of  the  earth,  is  probably  the  consequence  of  the 
other.     In  order  to  understand  their  relations  consider  a 


92         AN   INTRODUCTION   TO  ASTRONOMY'  [ch.  hi,  46 

pendulum  which  naturally  oscillates  in  seconds.  Suppose  it 
starts  from  rest  and  is  disturbed  by  a  small  periodic  force 
whose  period  is  two  thirds  of  a  second.  Presently  it  will  be 
moving,  not  like  an  undisturbed  pendulum,  but  with  one 
oscillation  in  two  thirds  of  a  second,  and  with  another 
oscillation  having  an  approximately  equal  magnitude,  in  its 
natural  period,  or  one  second. 

Euler  showed  about  1770  that  if  the  earth  were  absolutely 

rigid  the  natural  period  of  oscillation  of  its  pole  would  be  305 

'  days.     The  increase  of  period  to  428  day's  is  due  to  the  fact 

that  the  earth  yields  partially  to  disturbing  forces  (Art.  25). 

Many  parts  of  the  earth  have  experienced  wide  variations 
in  climate  during  geological  ages,  and  it  has  often  been  sug- 
gested that  these  great  changes  in  temperature  were  pro- 
duced by  the  wandering  of  its  poles.  There  are  no  known 
forces  which  could  produce  any  greater  variations  in  latitude 
than  those  which  have  been  considered,  and  there  is  not  the 
slightest  probability  that  the  earth's  poles  ever  have  been 
far  from  their  present  position  on  the  surface  of  the  earth. 

47.  Precession  of  the  Equinoxes  and  Nutation.  —  There 
is  one  more  phenomenon  to  be  considered  in  connection 
with  the  rotation  of  the  earth.  In  the  variation  of  latitude 
the  poles  of  the  earth  are  slightly  displaced  on  its  surface; 
now  the  changes  in  the  direction  of  its  axis  with  respect  to 
the  stars  are  under  consideration. 

The  axis  of  the  earth  can  be  changed  in  direction  only  by 
forces  exterior  to  itself.  The  only  important  exterior  forces 
to  which  the  earth  is  subject  are  the  attractions  of  the  moon 
and  sun.  If  the  earth  were  a  sphere,  these  bodies  would 
have  no  i  effect  upon  its  axis  of  rotation,  but  its  oblateness 
gives  rise  to  very  important  consequences. 

Let  0,  Fig.  31,  represent  a  point  on  the  equator  of  the 
oblate  earth,  and  suppose  the  moon  M  is  in  the  plane  of  the 
meridian  which  passes  through  0.  The  point  0  is  moving 
in  the  direction  OA  as  a  consequence  of  the  earth's  rotation. 
The  attraction  of  the  moon  for  a  particle  at  0  is  in  the  di- 


CH.  Ill,  47]      THE   MOTIONS   OF  THE   EARTH 


93 


rection.OJf.  By  the  resolution  of  forces  (the  inverse  of  the 
parallelogram  of  forces  law)  the  force  along  OM  can  be  re- 
solved in  two  others,  one  along  OE  and  the  other  along  the 
hne  OB  perpendicular  to  OE.  The  former  of  these  two 
forces  has  no  effect  on  the  rotation ;  the  latter  tends  to  move 


Fig.  31.  —  The  attraction   of  the   moon  for  the  earth's  equatorial  bulge 
causes  the  precession  of  the  equinoxes. 


the  particle  in  the  direction  OB,  and  this  tendency,  combined 
with  the  velocity  OA,  causes  it  to  move  in  the  direction  OC 
(the  change  is  greatly  exaggerated).  Therefore  the  direc- 
tion of  motion  of  0  is  changed;  that  is,  the  plane  of  the 
equator  is  changed. 

The  moon,  however,  attracts  every  particle  in  the  equatorial 
bulge  of  the  earth,  and  its  effects  vary  with  the  position  of 
the  particles.  It  can  be  shown  by  a  mathematical  discus- 
sion that  cannot  be  taken  up  here  that  the  combined  effect 
on  the  entire  bulge  is  to  change  the  plane  of  the  equator.  It 
is  evident  from  Fig.  31  that  the  effect  vanishes  when  the 
moon  is  in  the  plane  of  the,  earth's  equator.  Therefore  it 
is  natural  to  take  the  plane  of  the  moon's  orbit  as  a  plane  of 
reference.  These  two  planes  intersect  in  a  certain  line  whose 
position  changes  as  the  plane  of  the  earth's  equator  is  shifted. 
The  plane  of  the  earth's  equator  shifts  in  such  a  way  that 
the  angle  between  it  and  the  plane  of  the  moon's  orbit  is 
constant,  while  the  line  of  intersection  of  the  two  planes  ro- 


94         AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  47 

tates  in  the  direction  opposite  to  that  in  which  the  earth 
turns  on  its  axis. 

The  plane  in  which  the  sun  moves  is  called  the  plane  of 
the  ecliptic,  and  the  moon  is  always  near  this  plane.  For 
the  moment  neglect  its  departure  from  the  plane  of  the 
ecliptic.  Then  the  moon,  and  the  sun  similarly,  cause  the 
line  of  the  intersection  of  the  plane  of  the  earth's  equator 
and  the  plane  of  the  ecUptic,  called  the  line  of  the  equinoxes, 
to  rotate  in  the  direction  opposite  to  that  of  the  rotation  of 
the  earth.  This  is  the  precession  of  the  equinoxes,  four 
fifths  of  which  is  due  to  the  moon  and  one  fifth  of  which  is 
due  to  the  sun.  Since  the  axis  of  the  earth  is  perpendicular 
to  the  plane  of  its  equator,  the  point  in  the  sky  toward  which 
the  axis  is  directed  describes  a  circle  among  the  stars. 

The  mass  of  the  earth  is  so  great,  the  equatorial  bulge  is 
relatively  so  small,  and  the  forces  due  to  the  moon  and  sun 
are  so  feeble  that  the  precession  is  very  slow,  amounting  only 
to  50.2  seconds  of  arc  per  year,  from  which  it  follows  that 
the  line  of  the  equinoxes  will  make  a  complete  rotation  only 
after  more  than  25,800  years  have  passed. 

The  precession  of  the  equinoxes  was  discovered  by  Hip- 
parchus  about  120  B.C.  from  a  comparison  of  his  observa- 
tions with  those  made  by  earlier  astronomers,  but  the  cause 
of  it  was  not  known  until  it  was  explained  by  Newton,  in 
1686,  in  his  Principia.  The  theoretical  results  obtained  for 
the  precession  are  in  perfect  harmony  with  the  observations, 
and  the  weight  of  this  statement  will  be  appreciated  when 
it  is  remembered  that  the  calculations  depend  upon  the  size 
of  the  earth,  its  density,  the  distribution  of  mass  in  it,  the 
laws  of  motion,  the  rate  of  rotation  of  the  earth  and  its  oblate- 
ness,  the  distances  to  the  moon  and  sun,  their  apparent  mo- 
tions with  respect  to  the  earth,  and  the  law  of  gravitation. 

The  moon  does  not  move  exactly  in  the  plane  of  the 
ecliptic,  but  deviates  from  it  as  much  as  5  degrees,  and 
consequently  the  precession  which  it  produces  is  not  exactly 
with  respect  to  the  ecliptic.     This  circumstance  would  not 


CH.  Ill,  47]      THE   MOTIONS   OP   THE   EARTH  95 

be  particularly  important  if  it  were  not  for  the  further  fact 
that  the  plane  of  the  moon's  orbit  has  a  sort  of  precession 
with  respect  to  the  ecliptic,  completing  a  cycle  in  18.6  years. 
This  introduces  a  variation  in  the  character  of  the  precession 
which  is  periodic  with  the  same  period  of  18.6  years.  This 
variation  in  the  precession,  which  at  its  maximum  amounts 
to  9.2  seconds  of  arc,  is  called  the  nutation.  It  was  dis- 
covered by  the  great  English  astronomer  Bradley  from  ob- 
servations made  during  the  period  from  1727  to  1747.  The 
cause  of  it  was  first  explained  by  D'Alembert,  a  famous 
French  mathematician. 

V.  QUESTIONS 

1.  Which  of  the  proofs  of  the  rotation  of  the  earth  depend  upon 
the  laws  of  motion  ? 

2.  Give  three  practical  illustrations  (one  a  train  moving  around 
a  curve)  of  the  first  law  of  motion. 

3.  Give  three  illustrations  of  the  second  law  of  motion. 

4.  Why  is  the  kick  in  a  heavy  gun,  for  a  given  charge,  less  than 
in  a  hght  gun  ? 

5.  If  a  man  fixed  on  the  shore  pulls  a  boat  by  a  rope,  do  the 
interactions  not  violate  the  third  law  of  motion  ? 

6.  For  a  body  falUng  from  a  given  height,  in  what  latitude  will 
the  eastward  deviation  be  the  greatest  ? 

7.  For  what  latitude  will  the  rotation  of  the  Foucault  pendulum 
be  most  rapid,  and  where  would  the  experiment  fail  entirely  ? 

8.  In  what  latitude  will  the  easterly  (or  westerly)  deviation  of 
wind  or  water  currents  be  most  pronounced  ? 

9.  Is  it  easier  to  stop  a  large  or  small  wheel  of  the  same  mass 
rotating  at  the  same  rate  ? 

10.  If  a  wheel  rotating  without  friction  should  diminish  in  size, 
would  its  rate  of  rotation  be  a,ffected  ? 

11.  Are  boundaries  that  are  defined  by  latitudes  affected  by  the 
wabbling  of  the  earth's  axis  ?    By  the  precession  of  the  equinoxes  ? 

12.  Would  the  precession  be  faster  or  slower  if  the  earth  were 
more  oblate?    If  the  moon  were  nearer ?    If  the  earth  were  denser? 


96         AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  48 

II.  The  Revolution  of  the  Eabth 

48.  Relative  Motion  of  the  Earth  with  Respect  to  the 
Sun.  —  The  diurnal  motion  of  the  sun  is  so  obvious  that  the 
most  careless  observer  fully  understands  it.  But  it  is  not 
so  well  known  that  the  sun  has  an  apparent  eastward  motion 
among  the  stars  analogous  to  that  of  the  moon,  which  every 
one  has  noticed.  The  reason  that  people  are  not  so  famiUar 
with  the  apparent  motion  of  the  sun  is  that  stars  caimot 
be  observed  in  its  neighborhood  without  telescopic  aid,  and, 
besides,  it  moves  slowly.  However,  the  fact  that  it  ap- 
parently moves  can  be  estabUshed  without  the  use  of  optical 
instruments;    indeed,  it  was  known  in  very  ancient  times. 


Fig.  32.  —  The  hypothesis  that  the  sun  revolves  around  the  eirth  explains 
the  apparent  eastward  motion  of  the  sun  with  respect  to  the  stars. 

Suppose  on  a  given  date  certain  stars  are  seen  directly  south 
on  the  meridian  at  8  o'clock  at  night.  The  sun  is  therefore 
120°  west  of  the  star ;  or,  what  is  equivalent,  the  stars  in 
question  are  120°  east  of  the  sun.  A  month  later  at  8 
o'clock  at  night  the  observed  stars  will  be  found  to  be  30° 
west  of  the  meridian.  Since  at  that  time  in  the  evening  the 
sun  is  120°  west  of  the  meridian,  the  stars  are  120°  —  30° 
=  90°  east  of  the  sun.  That  is,  during  a  month  the  sun 
apparently  has  moved  30°  eastward  with  respect  to  the  stars. 
The  question  arises  whether  or  not  the  sun's  apparent 


CH.  Ill,  48]      THE   MOTIONS   OF   THE   EARTH  97 

motion  eastward  is  produced  by  its  actual  motion  around 
the  earljh.  It  will  be  shown  that  the  hypothesis  that  it 
actually  moves  around  the  earth  satisfies  all  the  data  so  far 
mentioned.  Suppose  E,  '¥'\g.  32,  represents  the  earth, 
assumed  fixed,  and  Si  the  position  of  the  sun  at  a  certain 
time.  As  seen  from  the  earth  it  Avill  appear  to  be  on  the  sky 
among  the  stars  at  Si'.  Suppose  that  at  the  end  of  25  days 
the  sun  has  moved  forward  in  a  path  around  the  earth  to 
the  position  S2 ;  it  will  then  appear  to  be  among  the  stars  at 
S'%.  That  is,  it  will  appear  to  have  moved  eastward  among 
the  stars  in  perfect  accordance  with  the  observations  of  its 
apparent  motion. 

It  will  now  be.  shown  that  the  same  observations  can  be 
satisfied  completely  by  the  hypothesis  that  the  earth  re- 


FlG.  33.  —  The  hypothesis  that  the  earth  revolves  around  the  sun  explains 
the  apparent  eastward  motion  of  the  sun  with  respect  to  the  stars. 

volves  around  the  sun.  Let  S,  Fig.  33,  represent  the  sun, 
assumed  fixed,  and  suppose  Ei  is  the  position  of  the  earth  at 
a  certain  time.  The  sun  will  appear  to  be  among  the  stars  at 
Si'.  Suppose  that  at  the  end  of  25  days  the  earth  has  moved 
forward  in  a  path  around  the  sun  to  E2. ;  the  sun  will  then 
appear  to  be  among  the  stars  at  &'.  That  is,  it  will  appear 
to  have  moved  eastward  among  the  stars  in  perfect  accord- 
ance with  the  observations  of  its  apparent  motion.  It  is 
noted  that  the  assumed  actual  motion  of  the  earth  is  in  the 
same  direction  as  the  sun's  apparent  motion ;  or,  to  explain 


98         AN   INTRODUCTION   TO  ASTRONOMY     [ch.  hi,  48 

the  apparent  motion  of  the  sun  by  the  motion  of  the  earth, 
the  earth  must  be  supposed  to  move  eastward  in  its  orbit. 

Since  all  the  data  satisfy  two  distinct  and  mutually  con- 
tradictory h3rpotheses,  new  data  must  be  employed  in  order 
to  determine  which  of  them  is  correct.  The  ancients  had 
no  facts  by  which  they  could  disprove  one  of  these  hypotheses 
and  estabhsh  the  truth  of  the  other. 

49.  Revolution  of  the  Earth  Proved  from  the  Laws  of 
Motion.  —  The  first  actual  proof  that  the  earth  revolves 
around  the  sun  was  based  on  the  laws  of  motion  in  1686, 
though  the  fact  was  generally  beheved  by  astronomers 
somewhat  earlier  (Art.  62).  It  must  be  confessed  at  once, 
however,  that  the  statement  requires  a  slight  correction  be- 
cause the  sun  and  earth  actually  revolve  around  the  center 
of  gravity  of  the  two  bodies,  which  is  very  near  the  center  of 
the  sun  because  of  the  sun's  relatively  enormous  mass. 

It  can  be  shown  by  measurements  that  have  no  connec- 
tion with  the  motion  of  the  sun  or  earth  that  the  volume  of 
the  sun  is  more  than  a  milUon  times  that  of  the  earth.  Hence, 
unless  it  is  extraordinarily  rare,  its  mass  is  much  greater 
than  that  of  the  earth.  In  view  of  the  fact  that  it  is  opaque, 
the  only  sensible  conclusion  is  that  it  has  an  appreciable 
density.  Hence,  in  the  motion  of  the  earth  and  sun  around 
their  common  center  of  gravity,  the  sun  is  nearly  fixed  while 
the  earth  moves  in  an  enormous  orbit. 

50.  Revolution  of  the  Earth  Proved  by  the  Aberration  of 
Light.  —  The  second  proof  that  the  earth  revolves  was 
made  in  1728  when  Bradley  discovered  what  is  known  as 
the  aberration  of  light.  This  proof  has  the  advantage  of 
depending  neither  on  an  assimaption  regarding  the  density 
of  the  sun  nor  on  the  laws  of  motion. 

Suppose  rain  falls  vertically  and  that  one  stands  still  in  it ; 
then  it  appears  to  him  that  it  comes  straight  down.  Suppose 
he  walks  rapidly  through  it ;  then  it  appears  to  fall  somewhat 
obliquely,  striking  him  in  the  face.  Suppose  he  rides  through 
it  rapidly ;  then  it  appears  to  descend  more  obliquely. 


CH.  Ill,  50]      THE   MOTIONS   OF   THE   EARTH 


99 


1      f      f 


A 

Fig.  6i.  —  Explanation  of 
the  aberration  of  light. 


In  order  to  get  at  the  matter  qualitatively  suppose  Ti 
Fig.  34,  is  a  tube  at  rest  which  is  to  be  placed  in  such  a 
position  that  drops  of  rain  shall  descend  through  it  without 
striking  the  sides.  Clearly  it  must  be  vertical.  Suppose  T^ 
is  a  tube  which  is  being  carried  to  the  right  with  moderate 
speed.  It  IS  evident  that  the  tube  must  be  tilted  shghtly 
in  the  direction  of  motion.  Suppose 
the  tube  Ts  is  being  transported  still 
more  rapidly;  it  must  be  given  a 
greater  deviation  from  the  vertical. 
The  distance  A3C3  is  the  distance  the 
tube  moves  while  the  drop  descends 
its  length.  Hence  AsCs  is  to  BiCs  as 
the  velocity  of  the  tube  is  to  the  ve- 
locity of  the  drops.  From  the  given 
velocity  of  the  rain  and  the  velocity 
of  the  tube  at  right  angles  to  the 
direction  of  the  rain,  the  angle  of  the  deviation  from  the 
vertical,  namely  A3B3C3,  can  be  computed. 

Now  suppose  Ught  from  a  distant  star  is  considered  in- 
stead of  faUing  rain,  and  let  the  tube  represent  a  telescope. 
All  the  relations  will  be  quahtatively  as  in  the  preceding 
case  because  the  velocity  of  light  is  not  infinite.  In  fact, 
it  has  been  found  by  experiments  on  the  €arth,  which  in  no 
way  depend  upon  astronomical  observations  or  theory,  that 
light  travels  in  a  vacuum  at  the  rate  of  186,330  miles  per 
second.  Hence,  if  the  earth  moves,  stars  should  appear 
displaced  in  the  direction  of  its  motion,  the  amount  of  the 
displacement  depending  upon  the  velocity  of  the  earth  and 
the  velocity  of  light.  Bradley  observed  such  displacements, 
at  one  time  of  the  year  in  one  direction  and  six  months  later, 
when  the  earth  was  on  the  other  side  of  its  orbit,  in  the 
opposite  direction.  The  maximum  displacement  of  a  star 
for  this  reason  is  20.47  seconds  of  arc  which,  at  the  present 
time,  is  very  easy  to  observe  because  measurements  of  po- 
sition are  now  accurate  to  one  hundredth  of  this  amount. 


100       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  50 

Moreover,  it  is  a  quantity  which  does  not  depend  on  the 
brightness  or  the  distance  of  the  star,  and  it  can  be  checked 
by  observing  as  many  stars  as  may  be  desired. 

The  aberration  of  light  not  only  proves  the  revolution  of 
the  earth,  but  its  amount  enables  the  astronomer  to  compute 
the  speed  with  which  the  earth  moves.  The  result  is  ac- 
curate to  within  about  one  tenth  of  one  per  cent.  Since 
the  earth's  period  around  the  sun  is  known,  this  result  gives 
the  circumference  of  the  earth's  orbit,  from  which  the  dis- 
tance from  the  earth  to  the  sun  can  be  computed.  The  dis- 
tance of  the  sun  as  found  in  this  way  agrees  very  closely 
with  that  found  by  other  methods. 

There  is,  similarly,  a  small  aberration  due  to  the  earth's 
rotation,  which,  for  a  point  on  the  earth's  equator,  amounts 
at  its  maximum  to  0.31  second  of  arc. 

51.  Revolution  of  the  Earth  Proved  by  the  Parallax  of 
the  Stars.  —  The  most  direct  method  of  testing  whether  or 
not  the  earth  moves  is  to  find  whether  the  direction  of  a 
star  is  the  same  when  observed  at  different  times  of  the 
year.  This  was  the  first  method  tried,  but  for  a  long  time 
it  failed  because  the  stars  are  exceedingly  remote.  Even 
with  all  the  resources  of  modem  instrumental  equipment 
fewer  than  100  stars  are  known  which  are  so  near  that  their 


-•/J 

Fia.  35.  —  The  parallax  of  A  is  the  angle  EiAEi. 

differences  in  direction  at  different  times  of  the  year  can  be 
measured  with  any  considerable  accuracy.  Yet  the  obser- 
vations succeed  in  a  considerable  number  of  cases  and  really 
prove  the  motion  of  the  earth  by  purely  geometrical  means. 
The  angular  difference  in  direction  of  a  star  as  seen  from 
two  points  on  the  earth's  orbit,  which,  in  the  direction  per- 
pendicular to  the  line  to  the  star,  are  separated  from  each 


CH.  Ill,  52]      THE   MOTIONS   OF   THE   EARTH  101 

other  by  the  distance  from  the  earth  to  the  sun,  is  the  par- 
allax of  the  star.  In  Fig.  35  let  S  represent  the  sun,  A  a 
star,  and  Ei  and  E2  two  positions  of  the  earth  such  that  the 
line  E1E2  is  perpendicular  to  SA  and  such  that  E1E2  equals 
^i-S.  Let  EiB  be  parallel  to  EiA.  Then,  by  definition, 
the  angle  AE^B  is  the  parallax  of  A.  This  angle  equals 
E1AE2.  Therefore  an  alternative  definition  of  the  parallax 
of  a  star  is  that  it  is  the  angle  subtended  by  the  radius  of 
the  earth's  orbit  as  seen  from  the  star. 

It  is  obvious  that  the  parallax  is  smaller  the  more  remote 
the  star.  The  nearest  known  star.  Alpha  Centauri,  in  the 
southern  heavens,  has  a  parallax  of  only  0.75  second  of  arc, 
from  which  it  can  be  shown  that  its  distance  is  275,000  times 
as  great  as  that  from  the  earth  to  the  sun,  or  about 
25,600,000,000,000  miles.  Suppose  a  point  of  light  is  seen 
first  with  one  eye  and  then  with  the  other.  If  its  distance 
from  the  observer  is  about  11  miles,  then  its  difference  in 
direction  as  seen  with  the  two  eyes  is  0.75  second  of  arc,  the 
parallax  of  Alpha  Centauri.  This  gives  an  idea  of  the 
difficulties  that  must  be  overcome  in  order  to  measure  the 
distance  of  even  the  nearest  star,  especially  when  it  is  re- 
called that  the  observations  must  be  extended  over  several 
months.  The  first  success  with  this  method  was  obtained 
by  Henderson  about  1840. 

52.  Revolution  of  the  Earth  Proved  by  the  Spectroscope. 
—  The  spectroscope  is  an  instrument  of  modem  invention 
which,  among  other  things,  enables  the  astronomer  to 
determine  whether  he  and  the  source  of  fight  he  may  be  ex- 
amining are  relatively  approaching  toward,  or  receding  from, 
each  other.  Moreover,  it  enables  him  to  measure  the 
speed  of  relative  approach  or  recession  irrespective  of  their 
distance  apart.     (Art.  226.) 

Consider  the  observation  of  a  star  A,  Fig.  36,  in  the  plane 
of  the  earth's  orbit  when  the  earth  is  at  Ei,  and  again  when 
it  is  at  E2.  In  the  first  position  the  earth  is  moving  toward 
the  star  at  the  rate  of  18.5  miles  per  second,  and  in  the  second 


102       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  52 

position  it  is  moving  away  from  the  star  at  the  same  rate. 
Since  in  the  case  of  many  stars  the  motion  can  be  determined 
to  within  one  tenth  of  a  mile  per  second,  the  observational 
difficulties  are  not  serious.     If  the  star  is  not  in  the  plane 

of  the  earth's  orbit,  a  cor- 

~/^   '\  ♦>«    rection  must  be  made  in 

I  ^  )'  order  to  find  what  fraction 

ft  of  the  earth's  motion  is 

Fig.  36.  —  Motion  of  the  earth  toward    toward   Or  from   the    star, 
and  from  a  star.  rm_  j.t_    j    ■      •    j  j 

The  method  is  independ- 
ent of  the  distance  of  the  star  and  can  be  applied  to  all 
stars  which  are  bright  enough  except  those  whose  directions 
from  the  sun  are  nearly  perpendicular  to  the  plane  of  the 
earth's  orbit. 

Since  1890  the  spectroscope  has  been  so  highly  perfected 
that  the  spectroscopic  proof  of  the  earth's  revolution  has  been 
made  with  thousands  of  stars.  This  method  gives  the 
earth's  speed,  and  therefore  the  circumference  of  its  orbit 
and  its  distance  from  the  sun.  It  should  be  stated,  however, 
that  the  motion  of  the  earth  was  long  ago  so  firmly  estab- 
hshed  that  it  has  not  been  considered  necessary  to  use  the 
spectroscope  to  give  additional  proof  of  it.  Rather,  it  has 
been  used  to  determine  how  the  stars  move  individually 
(Art.  273)  and  how  the  sun  moves  with  respect  to  them  as  a 
whole  (Art.  274).  In  order  to  obtain  the  motion  of  a  star 
with  respect  to  the  sun  it  is  sufficient  to  observe  it  when 
the  earth  is  at  E,  Fig.  36.  Then  correction  for  the  earth's 
motion  can  be  applied  to  the  observations  made  when  the 
earth  is  at  Ei  or  E^. 

53.  Shape  of  the  Earth's  Orbit.  —  It  has  been  tacitly 
assumed  so  far  that  the  earth's  orbit  is  a  circle  with  the  sun 
at  the  center.  If  this  assumption  were  true,  the  apparent 
diameter  of  the  sun  would  be  the  same  all  the  year  because 
the  earth's  distance  from  it  would  be  constant.  On  the 
other  hand,  if  the  sun  were  not  at  the  center  of  the  circle,  or 
if  the  orbit  were  not  a  circle,  the  apparent  size  of  the  sun 


CH.  Ill,  54]      THE   MOTIONS   OF   THE   EARTH 


103 


would  vary  with  changes  in  the  earth's  distance  from  it.  It 
is  clear  that  the  shape  of  the  earth's  orbit  can  easily  be 
established  by  observation  of  the  apparent  diameter  and 
position  of  the  sun. 

It  is  found  from  the^  changes  in  the  apparent  diameter  of 
the  sun  that  the  earth's  orbit  is  not  exactly  a  circle.  These 
changes  and  the  apparent  motion  of  the  sun  together  prove 
that  the  earth  moves  around  it  in  an  elliptical  orbit  which 
differs  only  a  little  from  a  circle.  An  elHpse  is  a  plane  curve 
such  that  the  sum  of  the  distances  from  two  fixed  points  in 
its  interior,  known  as  fod,  to  any  point  on  its  circumference 
is  always  the  same. 

In  Fig.  37,  E  represents  an  ellipse  and  F  and  F'  its  two 
foci.  The  definition  of  an  elhpse  suggests  a  convenient  way 
of  drawing  one.  Two 
pins  are  put  in  drawing 
paper  at  a  convenient 
distance  apart  and  a 
loop  of  thread  some- 
what longer  than  twice 
this  distance  is  placed 

over    them.        Then    a  Fig.  37. -An  ellipse. 

pencil  P  is  placed  inside  the  thread  and  the  curve  is  drawn, 
keeping  the  thread  taut.  The  curve  obtained  in  this  way  is 
obviously  an  ellipse  because  the  length  of  the  thread  is 
constant,  and  this  means  that  the  sum  of  the  distances 
from  F  and  F'  to  the  pencil  P  is  the  same  for  all  points  of 
the  curve. 

54.  Motion  of  the  Earth  in  Its  Orbit.  —  The  earth  moves 
in  its  orbit  around  the  sun  in  such  a  way  that  the  Une  drawn 
from  the  sun  to  the  earth  sweeps  over,  or  describes,  equal 
areas  in  equal  intervals  of  time.  Thus,  in  Fig.  38,  if  the 
three  shaded  areas  are  equal,  the  intervals  of  time  required 
for  the  earth  to  move  over  the  corresponding  arcs  of  its  orbit 
are  also  equal.  This  implies  that  the  earth  moves  fastest 
when  it  is  at  P,  the  point  nearest  the  sun,  and  slowest  when 


104       AN   INTRODUCTION   TO  ASTRONOMY     [ch.  iii,  54 


it  is  at  4,  the  point  farthest  from  the  sun.  The  former  is 
called  the  -perihelion  -point,  and  the  latter  the  aphelion  point. 
It  is  obvious  that  an  ellipse  may  be  very  nearly  roimd  or 
much  elongated.  The  extent  of  the  elongation  is  defined 
by  what  is  known  as  the  eccentricity,  which  is  the  ratio  CS 
divided  by  CP.  If  the  line  CS  is  very  short  for  a  given  hne 
CP,  the  eccentricity  is  small  and  the  ellipse  is  nearly  circular. 
In  fact,  a  circle  may  be  considered  as  being  an  ellipse  whose 
eccentricity  is  zero. 

The  eccentricity  of  the  earth's  orbit  is  very  slight,  being 
only  0.01677.     That  is,   the   distance  CS,   Fig.  38,  in  the 

case  of  the  earth's  orbit  is 
about  ^  of  CP.  Hence, 
if  the  earth's  orbit  were 
drawn  to  scale,  its  elonga- 
tion would  be  so  slight  that 
it  would  not  be  obvious  by 
simple  inspection. 

The  question  arises  as  to 
what  occupies  the  second 
focus  of  the  elliptical  orbit 
of  the  earth.  The  answer 
is  that  there  is  no  body 
there;  nor  is  it  absolutely 
fixed  in  position  because  the  earth's  orbit  is  continually 
modified  to  a  very  slight  extent  by  the  attractions  of  the 
other  planets. 

It  is  easy  to  see  how  the  earth  might  revolve  around  the 
sun  in  a  circle  if  it  were  started  with  the  right  velocity. 
But  it  is  not  so  easy  to  understand  how  it  can  revolve  in  an 
elliptical  orbit  with  the  sun  at  one  of  the  foci.  While  the 
matter  cannot  be  fully  explained  without  so'me  rather  for- 
midable mathematical  considerations,  it  can,  at  least,  be 
made  plausible  by  a  little  reflection.  Suppose  a  body  is  at 
P,  Fig.  38,  and  moving  in  the  direction  PT.  If  its  speed  is 
exactly  such  that  its  centrifugal  acceleration  balances  the 


Fig.  38.  —  The  earth  moves  so  that 
the  line  from  the  sun  to  the  earth 
sweeps  over  equal  areas  in  equal  inter- 
vals of  time. 


CH.  Ill,  55]      THE   MOTIONS   OP   THE   EARTH  105 

attraction  of  the  sun,  it  will  revolve  around  the  sun  in  a 
circle. 

But  suppose  the  initial  velocity  is  a  little  greater  than  that 
required  for  motion  in  a  circular  orbit.     In  this  case  the  sun's 
attraction   does   not   fully  counterbalance   the   centrifugal 
acceleration,  and  the  distance  of  the  body  from  the  sun 
increases.     Consider    the    situation    when    the    body    has 
moved  around  in  its  orbit  to  the  point  Q.    At  this  point  the 
centrifugal  acceleration  is  still  greater  than  the  attraction 
of  the  sun,  and  the  distance  of  the  body  from  the  sun  is 
increasing.     It  will  be  observed  that  the  sun's  attraction  no 
longer  acts  at  right  angles  to  the  direction  of  motion  of  the 
body,  but  that  it  tends  to  diminish  its  speed.     It  can  be 
shown  by  a  suitable  mathematical  discussion,  which  must  be 
omitted  here,  that  the  diminution  of  the  speed  of  the  body 
more  than  offsets  the  decreasing  attraction  of  the  sun  due  to 
the  increasing  distance  of  the  body,  and  that  in  elliptical 
orbits  a  time  comes  in  which  the  attraction  and  the  cen- 
trifugal acceleration    balance.     Suppose    this  takes    place 
when  the  body  is  at  fi.     Since  its  speed  is  still  being  dimin- 
ished by  the  attraction  of  the  sun  from  that  point  on,  the 
attraction  will  more  than  counterbalance  the.  centrifugal 
acceleration.     Eventually  at  A  the  distance  of  the  body  from 
the  sun  will  cease  to  increase.     That  is,  it  will  again  be  mov- 
ing at  right  angles  to  a  hne  joining  it  to  the  sun ;   but  its 
velocity  will  be  so  low  that  the  sun  will  pull  it  inside  of  a  cir- 
cular orbit  tangent  at  that  point.     It  will  then  proceed 
back  to  the  point  P,  its  velocity  increasing  as  it  decreases  in, 
distance  while  going  from  P  to  A.  ,  The  motion  out  from  the 
sun  and  back  again  is  analogous  to  that  of  a  ball  projected 
obHquely  upward  from  the  surface  of  the  earth ;    its  speed 
decreases  to  its  highest  point,  and  then  increases  again  as  it 
it  descends. 

55.  Inclination  of  the  Earth's  Orbit.  —  The  plane  of  the 
earth's  orbit  is  called  the  plane  of  the  ecliptic,  and  the  hne  in 
which  this  plane  intersects  the  sky  is  called  the  ecliptic.    In 


106       AN  INTRODUCTION  TO  ASTRONOMY    [ch.  hi,  55 

Fig.  39  it  is  the  circle  RAR'V.  The  plane  of  the  earth's 
equator  cuts  the  sky  in  a  circle  which  is  called  the  celestial 
equator.  In  the  figure  it  is  QAQ'V.  The  angle  between  the 
plane  of  the  equator  and  the  plane  of  the  ecliptic  is  23.5 
degrees.  This  angle  is  called  the  inclination  or  obliquity  of 
the  ecliptic. 

The  point  on  the  sky  pierced  by  a  line  drawn  perpendicular 
to  the  plane  of  the  ecliptic  is  called  the  pole  of  the  ecliptic, 


Fig.  39.  —  The  ecliptic,  celestial  equator,  and  celestial  pole. 


and  the  point  where  the  earth's  axis,  extended,  pierces  the 
sky  is  called  the  pole  of  the  equator  or,  simply,  the  celestial 
pole.  The  orbit  of  the  earth  is  so  very  small  in  comparison 
with  the  distance  to  the  sky  that  the  motion  of  the  earth  in 
its  orbit  has  no  sensible  effects  on  the  position  of  the  celestial 
pole  and  it  may  be  regarded  as  a  fixed  point.  In  Fig.  39, 
P'  is  the  pole  of  the  ecliptic  and  P  is  the  pole  of  the 
equator.  The  angle  between  these  lines  is  the  same  as  the 
angle  between  the  planes,  or  23.5  degrees. 

Now  consider  the  precession  of  the  equinoxes  (Art.  47). 


CH.  in,  56]      THE   MOTIONS   OF   THE   EARTH 


107 


The  pole  of  the  ecUptic  remains  fixed.  As  a  consequence  of 
the  precession  of  the  equinoxes  the  pole  P  describes  a  circle 
around  it  with  a 'radius  of  23.5  degrees,  and  the  direction  of 
the  motion  is  opposite  to  that  of  the  direction  of  the  motion  of 
the  earth  around  the  sun.  Or,  the  points  A  and  F,  which  are 
the  equinoxes,  continually  move  backward  along  the  ecliptic 
in  the  direction  opposite  to  that  of  the  revolution  of  the  earth. 

56.  Cause  of  the  Seasons.  —  Let  the  upper  part  of  the 
earth  E,  Fig.  39,  represent  its  north  pole.  When  the  earth 
is  at  Ex  its  north  pole  is  turned  away  from  the  sun  so  that 
it  is  in  continual  darkness;  but,  on  the  other  hand,  the 
south  pole  is  continually  illuminated.  At  this  time  of  the 
year  the  northern  hemisphere  has  its  winter  and  the  south- 
ern hemisphere  its  summer.  The  conditions  are  reversed 
when  the  earth  is  at  E^.  When  the  earth  is  at  E^,  the  plane 
of  its  equator  passes  through  the  sun,  and  it  is  the  spring 
season  in  the  northern  hemisphere.  Similarly,  when  the 
earth  is  at  Ei,  the  equator  also  passes  through  the  sun  and 
it  is  autumn  in  the  northern  hemisphere. 

Consider  a  point  in  a  medium  northern  latitude  when  the 
earth  is  at  Ei,  and  the  same  position  again  when  the  earth  is 
at  Ei.  At  El  the  sun's  rays, 
when  it  is  on  the  meridian, 
strike  the  surface  of  the  earth 
at  the  point  in  question  more 
obliquely  than  when  the  earth 
is  at  Ez.  Their  intensity  is, 
therefore,  less  in  the  former 
case  than  it  is  in  the  latter; 
for,  in  the  former,  the  rays 
whose  cross  section  is  PQ, 
Fig.  40,  are  spread  out  over 

the  distance  AB,  while  in  the  latter  they  extend  over  the 
smaller  distance  A'B.  This  fact,  and  the  variations  in  the 
number  of  hours  of  sunshine  per  day  (Art.  58),  cause  the 
changes  in  the  seasons. 


Fig. 


A    A'  a 

40.  —  Effects   of    obliquity    of 

sun'a  rays. 


108       AN    INTRODUCTION   TO  ASTRONOMY     [ch.  hi,  57 


57.  Relation  of  the  Position  of  the  Celestial  Pole  to  the 
Latitude  of  the  Observer.  —  In  order  to  make  clear  the 
climatic  effects  of  certaiii  additional  factors,  consider  the 
apparent  position  of  the  celestial  pole  as  seen  by  an  ob- 
server in  any  latitude.  Since  the  pole  is  the  place  where 
the  axis  of  the  earth,  extended,  pierces  the  sky,  it  is  obvious 
that,  if  an  observer  were  at  a  pole  of  the  earth,  the  celestial 
equator  would  be  on  his  horizon  and  the  celestial  pole  would 
be  at  his  zenith;  while,  if  he  were  on  the  equator  of  the 
earth,  the  celestial  equator  would  pass  through  his  zenith, 
and  the  celestial  poles  would  be  on  his  horizon,  north  and 
south. 

Consider  an  observer  at  0,  Fig.  41,  in  latitude  I  degrees 
north  of  the  equator.     The  line  P'P   points  toward  the 


Fig.  41.  —  The  altitude  of  the  celestial  pole  equals  the  .latitude  of   the 

observer. 

north  pole  of  the  sky.  Since  the  sky  is  extremely  far  away 
compared  to  the  dimensions  of  the  earth,  the  line  from  0 
to  the  celestial  pole  is  essentially  parallel  to  P'P.  The  angle 
between  the  plane  of  the  horizon  and  the  line  to  the  pole 
is  called  the  altitude  of  the  pole.     Since  ON  is  perpendicular 


CH.  Ill,  58]      THE   MOTIONS   OF   THE   EARTH 


109 


to  EO,  and  P'P  is  perpendicular  to  EQ,  it  follows  that  a 
equals  I,  or  the  altitude  of  the  pole  equals  the  latitude  of  the 
observer: 

Consider  also  the  altitude  of  the  equator  where  it  crosses 
the  meridian  directly  south  of  the  observer.  It  is  represented 
by  b  in  the  diagram.  It  easily  follows  that  6  =  90°  -  I, 
or  the  altitude  of  the  equator  where  it  crosses  the  meridian 
equals  90°  minus  the  latitude  of  the  observer. 

58.  The  Diumal  Circles  of  tha  Sun.  —  It  is  evident  from 
Fig.  39  that  when  the  earth  is  in  the  position  Ei,  the 
sun  is  seen  south  of  the  celestial  equator ;  when  the  earth  is 
at  E2  or  Ei,  the  sun  appears  to  be  on  the  celestial  equator ; 
and  when  the  earth  is  at  Ea,  the  sun  is  seen  north  of  the  ce- 
lestial equator.  If  the  equator  is  taken  as  the  line  of  refer- 
ence and  the  apparent  motion  of  the  sun  is  considered,  its 


Fig.  42.  —  Relation  of  ecliptic  and  celestial  equator. 


position  with  respect  to  the  equator  is  represented  in  Fig. 
42.  The  sun  appears  to  be  at  V  when  the  earth  is  at  E2, 
Fig.  39.  The  point  V  is  called  the  vernal  equinox,  and 
the  sun  has  this  position  on  or  within  one  day  of  March  21. 
The  sun  is  at  S,  called  the  summer  solstice,  when  the  earth  is 
at  E3,  Fig.  39,  and  it  is  in  this  position  about  June  21. 
The  sun  is  at  A,  called  the  autumnal  equinox,  when  the  earth 
is  at  Ei,  and  it  has  this  position  about  September  23.  Finally, 
the  sun  is  at  W,  which  is  called  the  winter  solstice,  when  the 
earth  is  at  Ei.  The  angle  between  the  echptic  and  the 
equator  at  V  and  A  is  23°. 5;  and  the  perpendicular  dis- 
tance between  the  equator  and  the  ecliptic  at  S  and  W  is 
23°.5.  From  these  relations  and  those  given  in  Art.  57  the 
diurnal  paths  of  the  sun  can  readily  be  constructed. 


no       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  iii,  58 

Suppose  the  observer  is  in  north  latitude  40°.  Let  0, 
Fig.  43,  represent  his  position,  and  suppose  his  horizon  is 
SWNE,  where  the  letters  stand  for  the  four  cardinal  points. 
Then  it  follows  from  the  relation  of  the  altitude  of  the  pole 

to  the  latitude  of  the  ob- 
server that  NP,  where  P 
represents  the  pole,  is  40°. 
Likewise  SQ,  where  Q  repre- 
sents the  place  at  which  the 
equator  crosses  the  meridian, 
is  50°.  The  equator  is  every- 
where 90  degrees  from  the 
pole  and  in  the  figure  is 
represented  by  the  circle 
'"■  QWQ'E. 

Suppose  the  sun  is  on  the 

Fig.  43.  —  Diurncil  circles  of  the  sun.  ,  ,    xr  a      -r^-       m 

equator  at  V  or  A,  Fig.  42. 
Since  it  takes  six  months  for  it  to  move  from  V  to  A,  its 
motion  in  one  day  is  very  small  and  may  be  neglected  in  the 
present  discussion.  Hence,  without  serious  error,  it  may  be 
supposed  that  the  sun  is  on  the  equator  all  day.  When  this 
is  the  case,  its  apparent  diurnal  path,  due  to  the  rotation 
of  the  earth,  is  EQWQ',  Fig.  43.  It  will  be  noticed  that 
it  rises  directly  in  the  east  and  sets  directly  in  the  west, 
being  exactly  half  the  time  above  the  horizon  and  half  the 
time  below  it.  This  is  true  whatever  the  latitude  of  the 
observer.  But  the  height  at  which  it  crosses  the  meridian 
depends,  of  course,  upon  the  latitude  of  the  observer,  and  is 
greater  the  nearer  he  is  to  the  earth's  equator. 

Suppose  now  that  it  is  June  21  and  that  the  sun  is  at  the 
summer  solstice  S,  Fig.  42.  It  is  then  23°. 5  north  of  the 
equator  and  will  have  essentially  this  distance  from  the 
equator  all  day.  The  diurnal  path  of  the  sun  in  this  case 
is  EiQiWiQi',  Fig.  43,  which  is  a  circle  parallel  to,  and  23°.5 
north  of,  the  equator.  In  this  case  the  sun  rises  north  of 
the  east  point  by  the  angle  EEi,  and  sets  an  equal  distance 


CH.  111,59]      THE   MOTIONS  OF  THE   EARTH  HI 

north  of  the  west  point.  Moreover,  it  is  more  than  half 
the  twenty-four  hours  above  the  horizon.  The  fact  that  its 
altitude  at  noon  is  23°. 5  greater  than  it  is  when  the  sun  is 
on  the  equator,  and  the  longer  time  from  sunrise  to  sunset, 
are  the  reasons  that  the  temperature  is  higher  in  the  summer 
than  in  the  spring  or  autumn.  It  is  obvious  from  Fig.  43 
that  the  length  of  the  day  from  sunrise  to  sunset  depends 
upon  the  latitude  of  the  observer,  being  greater  the  farther 
he  is  from  the  earth's  equator. 

When  the  sun  is  at  the  winter  solstice  W,  Fig.  42,  its 
diurnal  path  is  £'2621^2^2 '•  At  this  time  of  the  year  it  rises 
in  the  southeast,  crosses  the  meridian  at  a  low  altitude,  and 
sets  in  the  southwest.  The  time  during  which  it  is  above  the 
horizon  is  less  than  that  during  which  it  is  below  the  horizon, 
and  the  difference  in  the  two  intervals  depends  upon  the 
latitude  of  the  observer. 

59.  Hours  of  Sunlight  in  Different  Latitudes.  —  It  fol- 
lows from  Fig.  43  that  when  the  sun  is  north  of  the  celestial 
equator,  an  observer  north  of  the  earth's  equator  receives 
more  than  12  hours  of  sunlight  per  day ;  and  when  the  sun 
is  south  of  the  celestial  equator,  he  receives  less  than  12 
hours  of  sunUght  per  day.  It  might  be  suspected  that  the 
excess  at  one  time  exactly  balances  the  deficiency  at  the 
other.  This  suspicion  is  strengthened  by  the  obvious  fact 
that,  a  point  at  the  equator  receives  12  hours  of  sunhght 
per  day  every  day  in  the  year,  and  at  the  pole  the  sun 
shines  continuoi;sly  for  six  months  and  is  below  the  horizon 
for  six  months,  giving  the  same  total  number  of  hours  of 
sunshine  in  these  two  extreme  positions  on  the  earth.  The 
conclusion  is  correct,  for  it  can  be  shown  that  the  total 
number  of  hours  of  sunshine  in  a  year  is  the  same  at  all 
places  on  the  earth's  surface.  This  dbes  not,  of  course, 
mean  that  the  same  amount  of  sunshine  is  received  at  all 
places,  because  at  positions  near  the  poles  the  sun's  rays 
always  strike  the  surface  very  obhquely,  while  at  positions 
near  the  equator,  for  at  least  part  of  the  time  they  strike 


112       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi.  59 

the  surface  perpendicularly.  The  intensity  of  sunlight  at 
the  earth's  equator  when  the  sun  is  at  the  zenith  is  2.5 
times  its  maximum  intensity  at  the  earth's  poles ;  and  the 
amount  received  per  unit  area  on  the  equator  in  a  whole 
year  is  about  2.5  times  that  received  at  the  poles. 

If  the  obliquity  of  the  ecliptic  were  zero,  the  sun  would 
pass  every  day  through  the  zenith  of  an  observer  at  the 
earth's  equator ;  but  actually,  it  passes  through  the  zenith 
only  twice-  a  year.  Consequently,  the  effect  of  the  obUquity 
of  the  ecUptic  is  to  diminish  the  amount  of  heat  received  on 
the  earth's  equator.  Therefore  some  other  places  on  the 
earth,  which  are  obviously  the  poles,  must  receive  a  larger 
amount  than  they  would  if  the  equator  and  the  ecHptic 
were  coincident.  That  is,  the  obUquity  of  the  ecUptic 
causes  the  climate  to  vary  less  in  different  latitudes  than  it 
would  if  the  obliquity  were  zero. 

60.  Lag  of  the  Seasons.  —  From  the  astronomical  point 
of  view  March  21  and  September  23,  the  times  at  which  the 
sun  passes  the  two  equinoxes  are  corresponding  seasons. 
The  middle  of  the  summer  is  when  the  sun  is  at  the  summer 
solstice,  June  21,  and  the  middle  of  the  winter  when  it  is  at 
the  winter  solstice,  December  21.  But  from  the  climatic 
standpoint  March  21  and  September  23  are  not  correspond- 
ing seasons,  and  June  21  and  December  21  are  not  the 
middle  of  summer  and  winter  respectively.  The  climatic 
seasons  lag  behind  the  astronomical. 

The  cause  of  the  lag  of  the  seasons  is  very  simple.  On 
June  21  any  place  on  the  earth's  surface  north  of  the  Tropic 
of  Cancer  is  receiving  the  largest  amount  of  heat  it  gets  at 
any  time  in  the  year.  On  account  of  the  blanketing  effect 
of  the  atmosphere,  less  heat  is  radiated  than  is  received; 
hence  the  temperature  continues  to  rise.  But  after  that 
date  less  and  less  heat  is  received  as  day  succeeds  day ; 
on  the  other  hand,  more  is  radiated  daily,  for  the  hotter  a 
body  gets,  the  faster  it  radiates.  In  a  few  weeks  the  loss 
equals,  and  then  exceeds,  that  which  is  received,  after  which 


CH.  HI,  61]      THE   MOTIONS  OF  THE   EARTH 


113 


the  temperature  begins  to  fall.  The  same  reasoning  appUes 
for  all  the  other  seasons.  This  phenbmenon  is  quite  analo- 
gous to  the  familiar  fact  that  the  maximum  daily  tempera- 
ture normally  occurs  somewhat  after  noon. 

If  there  were  no  atmosphere  and  if  the  earth  radiated  heat 
as  fast  as  it  was  acquired,  there  would  be  no  lag  in  the 
seasons.  In  high  altitudes,  where  the  air  is  thin  and  dry, 
this  condition  is  nearly  realized  and  the  lag  of  the  seasons  is 
small,  though  the  phenomenon  is  very  much  disturbed  by  the 
great  air  currents  which  do  much  to  equahze  temperatures. 

61.  The  Effect  of  the  Eccentricity  of  the  Earth's  Orbit 
on  the  Seasons.  —  It  is  found  from  observations  of  the 


Fig.  44.  —  Because  of  the  eccentricity  of  the  earth's  orbit,  summers  in  the 
northern  hemisphere  are  longer  than  the  winters. 

apparent  diameter  of  the  sun  that  the  earth  is  at  its  peri- 
heUon  on  or  about  January  3,  and  at  its  aphelion  on  or  about 
July  4.  It  follows  from  the  way  the  earth  describes  its 
orbit,  as  explained  in  Art.  54,  that  the  time  required  for  it 
to  move  from  P  to  Q,  Fig.  44,  is  exactly  equal  to  that 
required  for  it  to  move  from  Q  to  P.  But  the  line  joining 
the  vernal  and  autumnal  equinoxes,  which  passes  through 
the  sun,  is  nearly  at  right  angles  to  the  Hne  joining  the 
perihehon  and  aphehon  points,  and  is  represented  by  VA, 
Fig. -44.  Since  the  area  swept  over  by  the  radius  from  the 
sun  to  the  earth,  while  the  earth  is  moving  over  the  arc 


114       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  61 

VQA,  is  greater  than  the  area  described  while  it  goes  over 
the  arc  APV,it  follows  that  the  interval  of  time  in  the  for- 
mer case  is  greater  than  that  in  th^  latter.  That  is,  since 
V  is  the  vernal  equinox,  the  summer  in,  the  northern  hemi- 
sphere is  longer  than  the  winter.  The  difference  in  length 
is  greatly  exaggerated  in  the  figure,  but  it  is  found  that  the 
interval  from  vernal  equinox  to  autumnal  equinox  is  actually 
about  186.25  days,  while  that  from  autumnal  equinox  to 
vernal  equinox  is  only  179  days.  The  difference  is,  there- 
fore, about  7.25  days. 

Since  the  siunmers  are  longer  than  the  winters  in  the 
northern  hemisphere  while  the  reverse  is  true  in  the  south- 
em  hemisphere,  it  might  be  supposed  that  points  in  corre- 
sponding latitudes  receive  more  heat  in  the  northern  hemi- 
sphere than  in  the  southern  hemisphere.  But  it  will  be 
noticed  from  Fig.  44  that,  although  the  summer  is  longer  in 
the  northern  hemisphere  than  it  is  in  the  southern,  the  earth 
is  then  farther  from  the  sun.  It  can  be  shown  from  a  dis- 
cussion of  the  way  in  which  the  earth's  distance  from  the 
sun  varies  and  from  the  rate  at  which  it  moves  at  differ- 
ent points  in  its  orbit,  that  the  longer  summer  season  in 
the  northern  hemisphere  is  exactly  counterbalanced  by  the 
greater  distance  the  earth  is  then  from  the  sun.  The  result 
is  that  points  in  corresponding  latitudes  north  and  south  of 
the  equator  receive  in  the  whole  year  exactly  the  same 
amount  of  Ught  and  heat  from  the  sun. 

There  is,  however,  a  difference  in  the  seasons  in  the  north- 
ern and  southern  hemispheres  which  depends  upon  the  ec- 
centricity of  the  earth's  orbit.  When  the  sun  is  north  of 
the  celestial  equator  so  that  its  rays  strike  the  surface  in 
northern  latitudes  most  nearly  perpendicularly,  a  condition 
that  tends  to  produce  high  temperatures,  the  greater  dis- 
tance of  the  sun  reduces  them  somewhat.  Therefore,  the 
temperature  does  not  rise  in  the  summer  so  high  as  it  would 
if  the  earth's  orbit  were  circular.  In  the  winter  time,  at 
the  same  place,  when  the  sun's  rays  strike  the  surface  slant- 


CH.  Ill,  62]      THE   MOTIONS   OF   THE   EARTH  115 

ingly,  the  earth  is  nearer  to  the  sun  than  the  average,  and 
consequently  the  temperature  does  not  fall  so  low  as  it  would 
if  the  eccentricity  of  the  earth's  orbit  were  zero.  The  re- 
sult is  that  the  seasonal  variations  in  the  northern  hemi- 
sphere are  less  extreme  than  they  would  be  if  the  earth's 
orbit  were  circular;  and,  for  the  opposite  reason,  in  the 
southern  hemisphere  they  are  more  extreme.  This  does 
not  mean  that  actually  there  are  greater  extremes  in  the 
temperature  south  of  the  equator  than  there  are  north  of  the 
equator.  The  larger  proportion  of  water  in  the  southern 
hemisphere,  which  tends  to  make  temperature  conditions 
uniform,  may  more  than  offset  the  effects  of  the  eccentricity 
of  the  earth's  orbit. 

The  attractions  of  the  other  planets  for  the  earth  change 
very  slowly  both  the  eccentricity  and  the  direction  of  the 
perihelion  of  the  earth's  orbit.  It  has  been  shown  by 
mathematical  discussions  of  these  influences  that  the  re- 
lation of  the  periheUon  to. the  line  of  the  equinoxes  will  be 
reversed  in  about  50,000  years.  In  fact,  there  is  a  cyclical 
change  in  these  relations  with  a  period  of  somewhat  more 
than  100,000  years.  It  was  suggested  by  James  Croll  that 
the  condition  of  long  winter  and  short  summer,  such  as 
now  prevails  in  the  southern  hemisphere,  especially  when  the 
eccentricity  of  the  earth's  orbit  was  greatest,  produced  the 
glaciation  which  large  portions  of  the  earth's  surface  are 
known  to  have  experienced  repeatedly  in  the  past.  This 
theory  has  now  been  abandoned  because;  on  other  grounds, 
it  is  extremely  improbable. 

62.  Historical  Sketch  of  the  Motions  of  the  Earth.  — 
The  history  of  the  theory  of  the  motion  of  the  earth  is  inti- 
mately associated  with  that  of  the  motions  of  the  planets, 
and  the  whole  problem  of  the  relations  of  the.  members  of 
the  solar  system  to  one  another  may  well  be  considered 
together. 

The  planets  are  readily  found  by  observations,  even 
without  telescopes,  to  be  moving  among  the  stars.     Theories 


116       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  62 

respecting  the  meanings  of  these  motions  date  back  to  the 
very  dawn  of  history.  Many  of  the  simpler  phenomena 
of  the  sun,  moon,  and  planets  had  been  carefully  observed 
by  the  Chaldeans  and  Egyptians,  but  it  remained  for  the 
brilliant  and  imaginative  Greeks  to  organize  and  generahze 
experience  and  to  develop  theories.  Thales  is  credited  with 
having  introduced  Egyptian  astronomy  into  Greece  more 
than  600  years  before  the  Christian  era.  The  Pythagoreans 
followed  a  century  later  and  made  important  contributions 
to  the  philosophy  of  the  science,  but  very  few  to  its  data. 
Their  success  was  due  to  the  weakness  of  their  method ;  for, 
not  being  too  much  hampered  by  the  facts  of  observation, 
they  gave  free  rein  to  their  imaginations  and  introduced 
numerous  ideas  into  a  budding  science  which,  though  often 
erroneous,  later  led  to  the  truth.  They  beUeved  that  the 
earth  was  round,  immovable,  at  the  center  of  the  universe, 
and  that  the  heavenly  bodies  moved  around  it  on  crystalline 
spheres. 

Following  the  Pythagoreans  came  Eudoxus  (409-356  B.C.), 
Aristotle  (384-322  b.c),  and  Aristarchus  (310-250  B.C.), 
who  were  much  more  scientific,  in  the  modern  sense  of 
the  term,  and  who  made  serious  attempts  to  secure  perfect 
agreement  between  the  observations  and  theory.  Aris- 
tarchus was  the  first  to  show  that  the  apparent  motions  of 
the  sun,  moon,  and  stars  could  be  explained  by  the  theory 
that  the  earth  rotates  on  its  axis  and  revolves  around  the 
sun.  Aristotle's  objection  was  that  if  this  theory  were  true 
the  stars  would  appear  to  be  in  different  directions  at  differ- 
ent times  of  the  year ;  the  reply  of  Aristarchus  was  that  the 
stars  were  infinitely  remote,  a  valid  answer  to  a  sensible 
criticism.  Aristarchus  was  a  member  of  the  Alexandrian 
school,  founded  by  Alexander  the  Great,  and  to  which  the 
geometer  Euclid  belonged.  His  astronomy  had  the  formal 
perfection  which  would  be  natural  in  a  school  where  geometry 
was  so  splendidly  systematized  that  it  has  required  almost 
no  modification  for  2000  years. 


CH.  Ill,  62]      THE   MOTIONS   OF   THE   EARTH  117 

The  rather  formal  astronomy  which  resulted  from  the 
influence  of  the  mathematics  of  Alexandria  was  succeeded 
by  an  epoch  in  which  the  greatest  care  was  taken  to  secure 
observations  of  the  highest  possible  precision.  Hipparchus 
(180-110  B.C.),  who  belonged  to  this  period,  is  universally 
conceded  to  have  been  the  greatest  astronomer  of  antiquity. 
His  observations  in  both  extent  and  accuracy  had  never  been 
approached  before  his  time,  nor  were  they  again  equaled 
until  the  time  of  the  Arab,  Albategnius  (850-929  a.d.). 
He  systematically  and  critically  compared  his  observations 
with  those  of  his  predecessors.  He  developed  trigonometry 
without  which  precise  astronomical  calculations  cannot  be 
made.  He  developed  an  ingenious  scheme  of  eccentrics 
and  epicycles  (which  will  be  explained  presently)  to  repre- 
sent the  motions  of  the  heavenly  bodies.  , 

Ptolemy  (100-170  a.d.)  was  the  first  astronomer  of  note 
after  Hipparchus,  and  the  last  important  astronomer  of  the 
Alexandrian  period.  From  his  time  until  that  of  Coper- 
nicus (1473-1543)  not  a  single  important  advance  was  made 
in  the  science  of  astronomy.  From  Pythagoras  to  Ptolemy 
was  700  years,  from  Ptolemy  to  Copernicus  was  1400  years, 
and  from  Copfernicus  to  the  present  time  is  400  years.  The 
work  of  Ptolemy,  which  is  preserved  in  the  Almagest  (i.e. 
The  Greatest  Composition),  was  the  crowning  achievement 
of  the  second  period,  and  that  of  Copernicus  was  the  first 
of  the  modem  period ;  or,  perhaps  it  would  be  more  accurate 
to  say  that  the  work  of  Copernicus  constituted  the  transition 
from  ancient  to  modern  astronomy,  which  was  really  begun 
by  Kepler  (1571-1630)  and  Galileo  (1564-1642). 

The  most  elaborate  theory  of  ancient  times  for  explaining 
the  motions  of  the  heavenly  bodies  was  due  to  Ptolemy. 
He  supposed  that  the  earth  was  a  fixed  sphere  situated  at 
the  center  of  the  universe.  He  supposed  that  the  sun  and 
moon  moved  around  the  earth  in  circles.  It  does  not  seem 
to  have  occurred  to  the  ancients  that  the  orbits  of  the  heavenly 
bodies  could  be  anything  but  circles,  which  were  supposed 


118       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  62 

to  be  perfect  curves.  In  order  to  explain  the  varying  dis- 
tances of  the  sun  and  moon,  which  were  proved  by  the  vari- 
ations in  thieir  apparent  diameters,  he  supposed  that  the 
earth  was  somewhat  out  of  the  centers  of  the  circles  in  which 
the  various  bodies  were  supposed  to  move  around  it.  It  is 
clear  that  such  motion,  called  eccentric  motion,  would  have 
considerable  similarity  to  motion  in  an  elhpse  around  a  body 
at  one  of  its  foci. 

Another  device  used  by  Ptolemy  for  the  purpose  of  ex-' 
plaining  the  motions  of  the  planets  was  the  epicycle.  In 
this  system  the  body  was  supposed  to  travel  with  uniform 
speed  along  a  small  circle,  the  epicycle,  whose  center  moved 
with  uniform  speed  along  a  large  circle,  the  deferent,  around 
the  earth.  By  carefully  adjusting  the  dimensions  and  in- 
cUnations  of  the  epicycle  and  the  deferent,  together  with 
the  rates  of  motion  along  them,  Ptolemy  succeeded  in  getting 
a  very  satisfactory  theory  for  the  motions  of  the  sun,  moon, 
and  planets  so  far  as  they  were  then  known. 

Copernicus  was  not  a  great,  or  even  a  skillful,  observer, 
but  he  devoted  many  years  of  his  life  to  the  study  of  the 
apparent  motions  of  the  heavenly  bodies  with  a  view  to 
discovering  their  real  motions.  The  invention  of  printing 
about  1450  had  made  accessible  the  writings  of  the  Greek 
philosophers,  and  Copernicus  gradually  became  convinced 
that  the  suggestion  that  the  sun  is  the  center,  and  that  the 
earth  both  rotates  on  its  axis  and  revolves  around  the  sun, 
explains  in  the  simplest  possible  way  all  the  observed  phe- 
nomena. It  must  be  insisted  that  Copernicus  had  no  rigorous 
proof  that  the  earth  revolved,  but  the  great  merit  of  his  work 
consisted  in  the  faithfulness  and  minute  care  with  which 
he  showed  that  the  heUocentric  theory  would  satisfy  the 
observation  as  well  as  the  geocentric  theory,  and  that  from 
the  standpoint  of  common  sense  it  was  much  more  plausible. 

The  immediate  successor  of  Copernicus  was  Tycho  Brahe 
(1546-1610),  who  rejected  the  heUocentric  theory  both  for 
theological  reasons  and  because  he  could  not  observe  any 


CH.  Ill,  62]      THE   MOTIONS  OF   THE   EARTH  119 

displacements  of  the  stars  due  to  the  annual  motion  of  the 
earth.  He  contributed  nothing  of  value  to  the  theory  of 
astronomy,  but  he  was  an  observer  of  tireless  industry  whose 
work  had  never  been  equaled  in  quality  or  quantity.  For 
example,  he  determined  the  length  of  the  year  correctly  to 
within  one  second  of  time. 

Between  the  time  of  Tycho  Brahe  and  that  of  Newton 
(1643-1727),  who  finally  laid  the  whole  foundation  for  me- 
chanics and  particularly  the  theory  of  motions  of  the  planets, 
there  Uved  two  great  astronomers,  GaUleo  (1564-1642)  and 
Kepler  (1571-1630),  who  by  work  in  quite  different  direc- 
tions led  to  the  complete  overthrow  of  the  Ptolemaic  theory 
of  eccentrics  and  epicycles.  These  two  men  had  almost  no 
characteristics  in  common.  Galileo  was  clear,  penetrating, 
brilUant ;  Kepler  was  mystical,  slow,  but  endowed  with  un- 
wearying industry.  Galileo,  whose  active  mind  turned  in 
many  directions,  invented  the  telescope  and  the  pendulum 
clock,  to  some  extent  anticipated  Newton  in  laying  the 
foundation  of  dynamics,  proved  that  light  and  heavy  bodies 
fall  at  the  same  rate,  covered  the  field  of  mathematical  and 
physical  science,  and  defended  the  hehocentric  theory  in  a 
matchless  manner  in  his  Dialogue  on  the  Two  Chief  Systems 
of  the  World.  Kepler  confined  his  attention  to  devising  a 
theory  to  account  for  the  apparent  motions  of  sun  and  planets, 
especially  as  measured  by  his  preceptor,  Tycho  Brahe.  With 
an  honesty  and  thoroughness  that  could  not  be  surpassed, 
he  tested  one  theory  after  another  and  found  them  unsatis- 
factory. Once  he  had  reduced  everything  to  harmony  ex- 
cept some  of  the  observations  of  Mars  by  Tycho  Brahe 
(of  course  without  a  telescope),  and  there  the  discrepancy 
was  below  the  limits  of  error  of  all  observers  except  Tycho 
Brahe.  Instead  of  ascribing  the  discrepancies  to  minute 
errors  by  Tycho  Brahe,  he  had  implicit  faith  in  the  absolute 
reliability  of  his  master  and  passed  on  to  the  consideration 
of  new  theories.  In  his  books  he  set  forth  the  complete 
record  of  his  successes  and  his  failures  with  a  childlike  candor 


120       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  hi,  62 

not  found  in  any  other  writer.  After  nearly  twenty  years 
of  computation  he  found  the  three  laws  of  planetary  motion 
(Art.  145)  which  paved  the  way  for  Newton.  Astronomy 
owes  much  to  the  thoroughness  of  Kepler. 

VI.  QUESTIONS 

1.  Note  carefully  the  position  of  any  conspicuous  star  at  8  p.m. 
and  verify  the  fact  that  in  a  month  it  will  be  30°  farther  west  at  the 
same  time  in  the  evening. 

2.  From  which  of  the  laws  of  motion  does  it  follow  that  two 
attracting  bodies  revolve  around  their  common  center  of  gravity  ? 

3.  What  are  the  fundamental  principles  on  which  each  of  the 
four  proofs  of  the  revolution  of  the  earth  depend  ?  How  many 
reaUy  independent  proofs  of  the  revolution  of  the  earth  are  there  ? 

4.  Which  of  the  proofs  of  the  revolution  of  the  earth  give  also 
the  size  of  its  orbit  ? 

5.  The  aberration  of  light  causes  a  star  apparently  to  describe  a 
small  curve  near  its  true  place ;  what  is  the  character  of  the  curve  if 
the  star  is  at  the  pole  of  the  ecliptic  ?  If  it  is  in  the  plane  of  the 
earth's  orbit? 

6.  Discuss  the'  questions  corresponding  to  question  5  for  the 
small  curve  described  as  a  consequence  of  the  parallax  of  a  star. 
Do  aberration  and  parallax  have  their  maxima  and  minima  at  the 
same  times,  or  are  their  phases  such  that  they  can  be  separated? 

7.  Discuss  the  cUmatic  conditions  if  the  day  were  twice  as  long 
as  it  is  at  present. 

8.  If  the  eccentricity  of  the  earth's  orbit  were  zero,  in  what 
respects  would  the  seasons  differ  from  those  which  we  have  now  ? 

9.  If  the  inclination  of  the  equator  to  the  eoUptic  were  zero,  in 
what  respects  would  the  seasons  differ  from  those  which  we  have  now  ? 

10.  Suppose  the  inchnation  of  the  equator  to  the  eohptic  were 
90°;  describe  the  phenomena  which  would  correspond  to  our  day 
and  to  our  seasons. 

11.  Draw  diagrams  giving  the  diurnal  circles  of  the  sun  when  the 
sun  is  at  an  equinox  and  both  solstices,  for  an  observer  at  the  earth's 
equator,  in  latitude  75°  north,  and  at  the  north  pole. 

12.  At  what  times  of  the  year  is  the  sun's  motion  northward  or 
southward  slowest  (see  Fig.  42)  ?  For  what  latitude  will  it  then  pass 
through  or  near  the  zenith  ?  This  place  will  thfen  have  its  highest 
temperature.  Compare  the  amount  of  heat  it  receives  with  that 
received  by  the  equator  during  an  equal  interval  when  the  sun  is 
near  the  equinox.    Which  will  have  the  higher  temperature.?  - 


CHAPTER  IV 
REFERENCE   POINTS   AND   LINES 

63.  Object  and  Character  of  Reference  Points  and 
Lines.  —  One  of  the  objects  at  which  astronomers  aim  is  a 
knowledge  of  the  motions  of  the  heavenly  bodies.  In  order 
fully  to  determine  their  motions  it  is  necessary  to  learn  how 
their  apparent  positions  change  with  the  time.  Another 
important  problem  of  the  astronomer  is  the  measurement  of 
the  distances  of  the  celestial  objects,  for  without  a  knowledge 
of  their  distances,  their  dimensions  and  many  other  of  their 
properties  cannot  be  determined".  In  order  to  measure  the 
distance  of  a  celestial  body  it  is  necessary  to  find  how  its 
apparent  direction  differs  as  seen  from  different  points  on 
the  earth's  surface  (Art.  123),  or  from  different  points  in  the 
the  earth's  orbit  (Art.  51).  For  both  of  these  problems  it  is 
obviously  important  to  have  a  precise  and  convenient  means 
of  describing  the  apparent  positions  of  the  heavenly  bodies. 

Not  only  are  systems  of  reference  points  and  lines  impor- 
tant for  certain  kinds  of  serious  astronomical  work,  but  they 
are  also  indispensable  to  those  who  wish  to  get  a  reasonable 
familiarity  with  the  wonders  of  the  sky.  Any  one  who  has 
traveled  and  noticed  the  stars  has  found  that  their  apparent 
positions  are  different  when  viewed  from  different  latitudes 
on  the  earth.  It  can  be  verified  by  any  one  on  a  single  clear 
evening  that  the  stars  apparently  move  during  the  night. 
And  if  the  sky  is  examined  at  the  same  time  of  night  on  dif- 
ferent dates  the  stars  will  be  found  to  occupy  different  places. 
That  is,  there  is  considerable  complexity  in  the  apparent 
motions  of  the  stars,  and  any  such  vague  directions  as  are 
ordinarily  made  to  suffice  for  describing  positions  on  the  earth 
would  be  absolutely  useless  when  appUed  to  the  heavens. 

121 


122       AN   INTRODUCTION   TO  ASTRONOMY     [ch.  iv,  63 

Although  the  celestial  bodies  differ  greatly  in  distance 
from  the  earth,  some  being  millions  of  times  as  far  away  as 
others,  they  all  seem  to  be  at  about  the  same  distance  on  a 
spherical  surface,  which  is  called  the  celestial  sphere.  In 
fact,  the  ancients  actually  assumed  that  the  stars  are  at- 
tached to  a  crystalline  sphere.  The  celestial  sphere  is  not  a 
sphere  at  any  particular  large  distance ;  it  is  an  imaginary 
surface  beyond  all  the  stars  and  on  which  they  are  all  pro- 
jected, at  such  an  enormous  distance  from  the  earth  that 
two  lines  drawn  toward  a  point  on  it  from  any  two  points 
on  the  earth,  or  from  any  two  points  on  the  earth's  orbit, 
are  so  nearly  parallel  that  their  convergence  can  never  be 
detected  with  any  instrument.  For  short,  it  is  said  to  be 
an  infinite  sphere. 

While  the  real  problem  giving  rise  to  reference  points  and 
lines  is  that  of  describing  accurately  and  concisely  the  direc- 
tions of  celestial  objects  from  the  observer,  its  solution  is 
equivalent  to  describing  their ,  apparent  positions  on  the 
celestial  sphere.  Since  it  is  much  easier  to  imagine  a  position 
on  a  sphere  than  it  is  to  think  of  the  direction  of  lines  radiat- 
ing from  its  center,  the  heavenly  bodies  are  located  in  direc- 
tion by  describing  their  projected  positions  on  the  celestial 
sphere.  Fortunately,  a  similar  problem  has  been  solved  ii 
locating  positions  on  the  surface  of  the  earth,  and  the  astro- 
nomical problem  is  treated  similarly. 

64.  The  Geographical  System.  —  Every  one  is  familiar 
with  the  method  of  locating  a  position  on  the  surface  of  the 
earth  by  giving  its  latitude  and  longitude.  Therefore  it  will 
be  sufficient  to  point  out  here  the  essential  elements  of  this 
process. 

The  geographical  lines  that  cover  the  earth  are  composed 
of  two  distinct  sets  which  have  quite  different  properties. 
The  first  set  consists  of  the  equator,  which  is  a  great  circle, 
and  the  parallels  of  latitude,  which  are  small  circles  parallel 
to  the  equator.  If  the  equator  is  defined  in  any  way,  the 
two  associated  poles,  which  are  90°  from  it,  are  also  uniquely 


CH.  IV,  65]     REFERENCE   POINTS  AND   LINES  123 

located.  Or,  if  there  is  any  natural  way  in  which  the  poles 
are  defined,  the  equator  is  itself  given.  In  the  case  of  the 
earth  the  poles  are  the  points  on  its  surface  at  the  ends  of 
its  axis  of  rotation,  and  these  points  consequently  have 
properties  not  possessed  by  any  others.  If  they  are  regarded 
as  being  defined  in  this  way,  the  equator  is  defined  as  the 
great  circle  90°  from  them. 

The  second  set  of  circles  on  the  surface  of  the  earth  con- 
sists of  great  circles,  called  meridians,  passing  through  the 
poles  and  cutting  the  equator  at  right  angles.  All  the 
meridians  are  similar  to  one  another,  and  a  convenient 
one  is  chosen  as  a  line  from  which  to  measure  longitudes. 
The  distances  from  the  fundamental  meridian  to  the  other 
meridians  are  given  in  degrees  and  are  most  conveniently 
measured  in  arcs  along  the  equator. 

The  fundamental  meridian  generally  used  ^s  a  standard 
is  that  one  which  passes  through  the  observatory  at  Green- 
wich, England.  However,  in  many  cases,  other  countries 
use  the  meridians  of  their  own  national  observatories.  For 
example,  in  the  United  States,  the  meridian  of  the  Naval 
Observatory  at  Washington  is  frequently  employed. 

In  order  to"  locate  uniquely  a  point  on  the  surface  of  the 
earth,  it  is  sufiicient  to  give  its  latitude,  which  is  the  angular 
distance  from  the  equator,  and  its  longitude,  which  is  the 
angular  distance  east  or  west  of  the  standard  meridian. 
These  distances  are  called  the  coordinates  of  the  point.  It 
is  customary  to  measure  the  longitude  either  east  or  west,  as 
may  be  necessary  in  order  that  it  shall  vuot  be  greater  than 
180°.  In  many  respects  it  would  be  simpler  if  longitude  were 
counted  from  the  fundamental  meridian  in  a  single  direction. 

65.  The  Horizon  System.  —  The  horizon,  which  separates 
the  visible  portion  of  the  sky  from  that  which  is  invisible,  is 
a  curve  that  cannot  escape  attention.  If  it  were  a  great 
circle,  it  might  be  taken  as  the  principal  circle  for  a  system  of 
coordinates  on  the  sky.  But  on  the  land  the  contour  of 
the  horizon  is  subject  to  the  numerous  irregularities  of  sur- 


124       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  iv,  65 


face,  and  on  the  sea  it  is  always  viewed  from  at  least  some 
small  altitude  above  the  surface  of  the  water.  For  this 
reason  it  is  called  the  sensible  horizon  to  distinguish  it  from 
the  astronomical  horizon,  which  will  be  defined  in  the  next 
paragraph. 

The  direction  defined  by  the  plmnb  hne  at  any  place 
is  perfectly  definite.  The  point  where  the  plvunb  line,  if 
extended  upward,  pierces  the  celestial  sphere  is  called  the 
zenith,  and  the  opposite  point  is  called  the  nadir.  These  two 
points  will  be  taken  as  poles  of  the  first  set  of  coordinates  in 
the  horizon  system,  and  the  horizon  is  defined  as  the  great 
circle  on  the  celestial  sphere  90°  from  the  zenith.  The  small 
circles  parallel  to  the  horizon  are  called  altitude  circles  or, 
sometimes,  almucantars. 

The  second  set  of  circles  in  the  horizon  system  consists  of 
the  great  circles  which  pass  through  the  zenith  and  the 
nadir  and  cut  the  horizon  at  right  angles.  They  are  called 
vertical  circles.  The  fundamental  vertical  circle  from  which 
distances  along  the  horizon  are  measured  is  that  one  which 
2  passes  through  the  pole  of 

the  sky ;  that  is,  the  point 
where  the  axis  of  the  earth, 
prolonged,  cuts  the  celestial 
sphere,  and  it  is  called  the 
meridian. 

The  coordinates  of  a  point 

in  the  horizon  system  are  (a) 

the  angular  distance  above  or 

below  the  horizon,  which  is 

called  altitude,  and   (b)   the 

angular  distance  west  from 

the  south  point   along    the 

horizon  to  the  place  where  the  vertical  circle  through  the 

object  crosses  the  horizon.     This  is  called  the  azimuth  of 

the  object. 

In  Fig.  45,  0  represents  the  position  of  the  observer, 


/^ 

n'"i'"i3 

Pi-' 

-f-;i^-, 

-  yn 

Fig.  45.  —  The  horizon  system. 


CH.  IV,  68]     REFERENCE   POINTS  AND   LINES  125 

SWNE  his  horizon,  and  Z  his  zenith.  The  point  where  the 
earth's  axis  pierces  the  sky  is  perfectly  definite  and  is  repre- 
sented by  P  in  the  diagram.  The  vertical  circle  which  passes 
through  Z  and  P  is  the  meridian.  The  points  at  which  the 
meHdian  cuts  the  horizon  are  the  north  and  south  points. 
The  north  point,  for  positions  in  the  northern  hemisphere 
of  the  earth,  is  the  one  nearest  the  pole  P.  In  this  way  the 
cardinal  points  are  uniquely  defined. 

Consider  a  star  at  A.  Its  altijiude  is  BA,  which,  in  this 
case,  is  about  40°,  and  its  azimuth  is  SWNEB,  which,  in 
this  case,  is  about  300°.  It  is,  of  course,  understood  that 
the  object  might  be  below  the  horizon  and  the  azimuth 
might  be  anything  from  zero  to  360°.  When  the  object  is 
above  the  horizon,  its  altitude  is  considered  as  being  positive, 
and  when  below,  as  being  negative. 

66.  The  Equator  System.  —  The  poles  of  the  sky  have 
been  defined  as  the  points  where  the  earth's  axis  prolonged 
intersects  the  celestial  sphere.  It  might  be  supposed  at 
first  that  these  would  not  be  conspicuous  points  because  the 
earth's  axis  is  a  line  which  of  course  cannot  be  seen.  But 
the  rotation  of  the  earth  causes  an  apparent  motion  of 
the  stars  around  the  pole  of  "the  sky.  Consequently,  an 
equally  good  definition  of  the  poles  is  that  they  are  the 
comnaon  centers  of  the  diurnal  circles  of  the  stars.  That 
pole  which  is  visible  from  the  position  of  an  observer  is  a 
point  no  less  conspicuous  than  the  zenith. 

The  celestial  equator  is  a  great  circle  90°  from  the  poles 
of  the  sky.  An  alternative  definition  is  that  the  celestial 
equator  is  the  great  circle  in  which  the  plane  of  the  earth's 
equator  intersects  the  celestial  sphere.  The  small  circles 
parallel  to  the  celestial  equator  are  called  declination  circles. 

The  second  set  of  circles  in  the  equatorial  system  consists 
of  those  which  pass  through  the  poles  and  are  perpendicular 
to  the  celestial  equator.  They  are  called  hour  circles.  The 
fundamental  hour  circle,  called  the  equinoctial  colure,  from 
which  all  others  are  measured,  is  that  one  which  passes 


126       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  iv,  66 

through  the  vernal  equinox,  that  is,  the  place  at  which  the 
sun  in  its  apparent  annual  motion  around  the  sky  crosses 
the  celestial  equator  from  south  to  north. 

The  coordinates  in  the  equator  system  are  (a)  the  angular 
distance  north  or  south  of  the  celestial  equator,  which  is  called 
declination,  and  (b)  the  angular  distance  eastward  from  the 
vernal  equinox  along  the  equator  to  the  point  where  the 
hour  circle  through  the  object  crosses  the  equator.  This 
distance  is  called  right  ascension.  The  direction  eastward  is 
defined  as  that  in  which  the  sun  moves  in  its  apparent 
motion  among  the  stars. 

In  Fig.  46,  let  0  represent  the  position  of  the  observer, 
NESW  his  horizon,  PNQ'SQ  his  meridian.     Suppose  the 

star  is  at  A  and  that  the  ver- 
nal equinox  is  at  V.  Then  the 
declination  of  the  star  is  the 
arc  CA  and  its  right  ascension 
is  VQC.  In  this  case  the  dec- 
;j/vlination  is 'about  40°  and  the 
right  ascension  is  about  75°. 
It  is  not  customary  to  express 
the  right  ascension  in  degrees, 
but  to  give  it  in  hours,  where 
an  hour  equals  15°.      In  the 

FiQ.  46.  —  The  equator  system.  ,  , ,  .    ,  , 

present  case  the  right  ascen- 
sion of  A  is,  therefore,  about  5  hours. 

It  is  easy  to  understand  why  it  is  convenient  to  count 
right  ascension  in  hours.  The  sky  has  an  apparent  motion 
westward  because  of  the  earth's  actual  rotation  eastward, 
and  it  makes  a  complete  circuit  of  360°  in  24  hours.  There- 
fore it  apparently  moves  westward  15°  in  one  hour.  It 
follows  that  a  simple  method  of  finding  the  right  ascension 
of  an  object  is  to  note  when"  the  vernal  equinox  crosses  the 
meridian  and  to  measure  the  time  which  elapses  before  the 
object  is  observed  to  cross  the  meridian.  The  interval  of 
time  is  its  right  ascension  expressed  in  hours. 


CH.  IV,  68]     REFERENCE   POINTS  AND   LINES  127 

67.  The  Ecliptic  System.  *—  The  third  system  which  is 
employed  in  astronomy,  but  much  less  frequently  than  the 
other  two,  is  known  as  the  ecUptic  system  because  the  funda- 
mental circle  in  its  first  set  is  the  ecliptic.  The  ecliptic  is 
the  great  circle  on  the  celestial  sphere  traced  out  by  the  sun 
in  its  apparent  annual  motion  around  the  sky.  The  points 
on  the  celestial  sphere  90°  from  the  ecliptic  are  the  poles  of 
the  ecUptic.  The  small  circles  parallel  to  the  ecliptic  are 
called  parallels  of  latitude.  The  great  circles  which  cross 
the  ecliptic  at  right  angles  are  called  longitude  circles. 

The  coordinates  in  the  ecliptic  system  are  the  angular  dis- 
tance north  or  south  of  the  ecliptic,  which  is  called  latitude, 

and    the    distance    eastward  , 

from  the  vernal  equinox  along  xf^^Cl'  '■        ~"^\ 

the  ecUptic  to  the  point  where  /\  ■■■'■.  '•                \f 

the   longitude   circle    through  /      \    1  ';•.'■,                \ 

the  object  intersects  the  eclip-      /    Vrx'f^y ■---,( 

tic,  which  is  called  longitude.       =^- "'M;;;°  V',         ^^ 

In  Fig.  47,  0  represents  the  \^          ^^^- — r\       , 

position  of  the  observer  and  \                \\'      '      / 

QEQ'W  the  celestial  equator.  N^            \  N.     \/ 

Suppose  that  at  the  time  in  ^^^~-___3J-^^'^^ 

question  the  vernal  equinox  is  ^      ,^      ^,       ,'  . 

^     ^,  ,      ,  ,  ,  ,        Fig.  47.  —  The  ecliptic  system. 

at  V  and  that  the  autumnal  ^ 

equinox  is  at  ^.     Then,  since  the  angle  between  the  echptic 

and  the  equator  is  23°.5,  the  position  of  the  ecUptic  is 

AX'VX. 

68.  Comparison  of  the  Systems  of  Coordinates.  —  All 
three  of  the  systems  of  coordinates  are  geometrically  Uke  the 
one  used  in  geography ;  but  there  are  important  differences 
in  the  way  in  which  they  arise  and  in  the  purposes  for  which 
their  use  is  convenient. 

The  honzon  system  depends  upon  the  position  of  the 
observer  and  the  direction  of  his  plumb  line.  It  always 
has  the  same  relation  to  him,  and  if  he  travels  he  takes  it 
with  fifen.     The  equator  system  is  defined  by  the  apparent 


128       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  iv,  68 

rotation  of  the  sky,  which  is  due,  of  course,  to  the  actual 
rotation  of  the  earth,  and  it  is  altogether  independent  of 
the  position  of  the  observer.  The  ecUptic  system  is  defined 
by  the  apparent  motion  of  the  sun  around  the  sky  and  also 
is  independent  of  the  position  of  the  observer. 

Since  the  horizon  system  depends  upon  the  position  of 
the  observer,  the  altitude  and  azimuth  of  an  object  do  not 
really  locate  it  unless  the  place  of  the  observer  is  given. 
Since  the  stars  have  diurnal  motions  across  the  sky,  the  time 
of  day  must  also  be  given ;  and  since  different  stars  cross  the 
meridian  at  different  times  on  succeeding  days,  it  follows 
that  the  day  of  the  year  must  also  be  given.  The  incon- 
venience of  the  horizon  system  arises  from  the  fact  that  its 
circles  are  not  fixed  on  the  sky.  Yet  it  is  important  for  the 
observer  because  the  horizon  is  approximately  the  boundary 
which  separates  the  visible  from  the  invisible  portion  of 
the  sky. 

In  the  equator  system  the  reference  points  and  lines  are 
fixed  with  respect  to  the  stars.  This  statement,  however, 
requires  two  slight  corrections.  In  the  first  place,  the 
earth's  equator,  and  therefore  the  celestial  equator,  is  subject 
to  precession  (Art.  47).  In  the  second  place,  the  stars  have 
very  small  motions  with  reference  to  one  another  which 
become  appreciable  in  work  of  extreme  precision,  generally 
in  the  course  of  a  few  years.  But  in  the  present. connection 
these  motions  will  be  neglected  and  the  equator  coordinates 
will  be  considered  as  being  absolutely  fixed  with  respect 
to  the  stars.  With  this  understanding  the  apparent  position 
of  an  object  is  fully  defined  if  its  right  ascension  and  dechna- 
tion  are  given.  The  reference  points  and  lines  of  the  ecliptic 
system  also  have  the  desirable  quality  of  being  fixed  with 
respect  to  the  stars. 

From  what  has  been  said  it  might  be  inferred  that  the 
equator  and  ecliptic  systems  are  equally  convenient,  but 
such  is  by  no  means  the  case.  The  equator  always  crosses 
the  meridian  at  an  altitude  which  is  equal  to  90°  minus  the 


CH.  IV,  68]     REFERENCE   POINTS  AND   LINES 


129 


latitude  of  the  observer  (Art.  57)  and  always  passes  through 
the  east  and  west  points  of  the  horizon.  Consequently,  all 
objects  having  the  saine  dechnation  cross  the  meridian  at  the 
same  altitude.  Suppose,  for  example,  that  the  observer  is  in 
latitude  40°  north.  Then  the  equator  crosses  his  meridian 
at  an  altitude  of  50°.  If  he  observes  that  a  star  crosses 
the  meridian  at  an  altitude  of  60°,  he  knows  that  it  is  10° 
north  of  the  celestial  equator,  or  that  its  declination  is  10° ; 
and  by  noting  the  time  that  has  elapsed  from  the  time  of 
the  passage  of  the  vernal  equinox  across  the  meridian  to  the 
passage  of  the  star,  he  has  its  right  ascension.  Nothing 
could  be  simpler  than  getting  the  coordinates  of  an  object  in 
the  equator  system. 

Now  consider  the  ecliptic  system.     Suppose  V,  in  Fig.  48, 
represents  the  position  of  the  vernal  equinox  on  a  certain 


Fig.  48. 


Fig.  49. 


Equator  and  ecliptic. 


date  and  time  of  day.  Then  the  pole  of  the  echptic  XVX'A 
is  at  R  and  the  ecliptic  crosses  the  meridian  below  the 
equator.  In  this  case  the  star  might  have  north  celestial 
latitude  and  be  on  the  meridian  south  of  the  equator.  Twelve 
hours  later  the  vernal  equinox  has  apparently  rotated  west- 
ward with  the  sky  to  the  point  V,  Fig.  49.  The  pole  of  the 
ecliptic  has  gone  around  the  pole  P  to  the  point  R,  and  the 
ecliptic  crosses  the  meridian  north  of  the  equator.     It  is 


130       AN   INTRODUCTION   TO  ASTRONOMY     [ch.  iv,  68 

clear  from  Figs.  48  and  49  that  the  position  of  the  ecliptic 
with  respect  to  the  horizon  system  changes  continually  with 
the  apparent  rotation  of  the  sky.  It  follows  that  for  most 
purposes  the  ecliptic  system  is  not  convenient.  Its  use 
in  astronomy  is  limited  almost  entirely  to  describing  the 
position  of  the  sun,  which  is  always  on  the  ecliptic,  and 
the  positions  of  the  moon  and  planets,  which  are  always 
near  it. 

69.  Finding  the  Altitude  and  Azimuth  when  the  Right 
Ascension,  Declination,  and  Time  are  Given.  —  Suppose 
the  right  ascension  and  declination  of  a  star  are  given  and 
that  its  altitude  and  azimuth  are  desired.  It  is  necessary 
also  to  have  given  the  latitude  of  the  observer,  the  time  of 
day,  and  the  time  of  year,  because  the  altitude  and  azimuth 
depend  on  these  quantities.  Most  of  the  difficulty  of  the 
problem  arises  from  the  fact  that  the  vernal  equinox  has  a 
diurnal  motion  around  the  sky  and  that  it  is  a  point  which 
is  not  easily  located.  By  computing  the  right  ascension  of 
the  sun  at  the  date  in  question,  direct  use  of  the  vernal 
equinox  may  be  avoided.  It  has  been  found  convenient  to 
solve  the  problem  in  four  distinct  steps. 

Step  1 .  The  right  ascension  of  the  sun  on  the  date  in  question. 
—  It  has  been  found  by  observation  that  the  sun  passes 
the  vernal  equinox  March  21.  (The  date  may  vary  a  day 
because  of  the  leap  year,  but  it  will  be  sufficiently  accurate 
for  the  present  purposes  to  use  March  21  for  all  cases.)  In 
a  year  the  sun  moves  around  the  sky  24  hours  in  right  ascen- 
sion, or  at  the  rate  of  two  hours  a  month.  Although  the 
rate  of  apparent  motion  of  the  sun  is  not  perfectly  uniform, 
the  variations  from  it  are  small  and  will  be  neglected  in  the 
present  connection.  It  follows  from  these  facts  that  the 
right  ascension  of  the  sun  on  any  date  may  be  found  by 
counting  the  number  of  months  from  March  21  to  the  date 
in  question  and  multiplying  the  result  by  two.  For  ex- 
ample, October  6  is  6.5  months  from  March  21,  and  the 
right  ascension  of  the  sun  on  this  date  is,  therefore,  13  hours. 


CB,  IV,  69]     REFERENCE   POINTS  AND   LINES  131 

Ste-p  2.  The  right  ascension  of  the  meridian  at  the  given 
time  of  day  on  the  date  in  question.  —  Suppose  the  right 
ascension  of  the  sun  has  been  determined  by  Step  1.  Since 
the  sun  moves  360°  in  365  days,  or  only  one  degree  per  day, 
its  motion  during  one  day  may  be  neglected.  Suppose,  for 
example,  that  it  is  8  o'clock  at  night.  Then  the  sun  is  8 
■  hours  west  of  the  meridian  at 
the  position  indicated  in  Fig.  oy''^  """^ 

50.     Since  right  ascension  is         /\     "■•  >r 

counted  eastward  and  the  right       /    \       '"  \  \ 

ascension  of  the  sun  is  known,      /        \ --v"?...  \ 

the  right  ascension  of  Q  may    S''         \     o/  '\  "j« 

be  found  by  adding  the  num-     \^~- — ^  \,''       ^fp.,-^ 

ber  of  hours  from  the  sun  to  Q      \  ^\^        \       / 

to  the  right  ascension  of  the        \  n^^    \  / 

sun.'    If  the  right  ascension  of  \^^  Jy^ 

the  sun  is  13  hours  and  the 

time  of  the  day  is  8  p.m.,  the  ^'°-  ™' ~the  meridian "''°''°°  °^ 

right  ascension  of  the  meridian 

is  13  +  8  =  21  hours.     The  general  rule  is,  the  right  ascension 

of  the  meridian  is  obtained  by  adding  to  the  right  ascension 

of  the  sun  the  number  of  hours  after  noon. 

Btep  3.  The  hour  angle  of  the  object.  —  Wherever  the  object 
may  be,  a  certain  hour  circle  passes  through  it  and  crosses 
the  equator  at  some  point.  The  distance  from  the  meridian 
along  the  equator  to  this  point  is  called  the  hour  angle  of 
the  object.  The  hour  angle  is  counted  either  east  or  west 
as  may  be  necessary  in  order  that  the  resulting  number 
shall  not  exceed  12. 

Suppose  the  right  ascension  of  the  meridian  has  been 
found  by  Step  2.  The  hour  angle  of  the  star  is  the  difference 
between  its  right  ascension,  which  is  one  of  the  quantities 
given  in  the  problem,  and  the  right  ascension  of  the  meridian. 
If  the  right  ascension  of  the  star  is  greater  than  that  of  the 
meridian,  its  hour  angle  is  east,  and  if  it  is  less  than  that  of  the 
meridian,  its  hour  angle  is  west.    There  is  one  case  which. 


132       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  iv,  69 


in  a  way,  is  an  exception  to  this  statement.  Suppose  the 
right  ascension  of  the  meridian  is  22  hours  and  the  right 
ascension  of  the  star  is  2  hours.  According  to  the  rule  the 
star  is  20  hours  west,  which,  of  course,  is  the  same  as  4  hours 
east.  But  its  right  ascension  of  2  hours  may  be  considered 
as  being  a  right  ascension  of  26  hours,  just  as  2  o'clock  in  the 
afternoon  can  be  equally  well  called  14  o'clock.  When  its 
right  ascension  is  called  26  hours,  the  rule  leads  directly  to 
the  result  that  the  hour  angle  is  4  hours  east. 

Step  4-  Application  of  the  declination  and  estimation  of 
the  altitude  and  azimuth.  —  In  order  to  make  the  last  step 
clear,  consider  a  special  example.     Suppose  the  hour  angle 

of  the  object  has  been  found 
by  Step  3  to  be  7  hours  east. 
This  locates  the  point  C,  Fig. 
51.  Therefore  the  star  is 
somewhere  on  the  hour  circle 
PCP'.  The  given  decUna- 
tion  determines  where  the 
star  is  on  the  circle.  Sup- 
pose, for  example,  that  the 
object  is  35°  north.  In  order 
to  locate  it,  it  is  only  neces- 
sary to  measure  off  35° 
from  C  along  the  circle  CP 
Hence  the  star  is  at  A. 
Now  draw  a  vertical  circle  from  Z  through  A  to  the  horizon 
at  B.  The  altitude  is  BA  and  the  azimuth  is  SWNB. 
These  distanqes  can  be  computed  by  spherical  trigonometry, 
but  they  may  be  estimated  closely  enough  for  present 
purposes.  In  this  problem  the  altitude  is  about  12°  and  the 
azimuth  is  about  230°.  Whatever  the  data  may  be  which 
are  supplied  by  the  problem,  the  method  of  procedure  is 
always  that  which  has  been  given  in  the  present  case. 

70.  Illustrative  Example  for  Finding  Altitude  and 
Azimuth.  —  In  order  to  illustrate  fully  the  processes  that 


Fig.  51.  — Application  of  the  declina- 
tion in  finding  the  position  of  a  star. 


CH.  IV,  71]     REFERENCE   POINTS  AND   LlNES  133 

have  been  explained  in  Art.  69,  an  actual  problem  will  be 
solved.  Suppose  the  observer  is  in  latitude  40°  north.  The 
altitude  of  the  pole  P,  Fig.  52,  as  seen  from  his  position,  will 
be  40°,  and  the  point  Q,  where  the  equator  crosses  the 
meridian,    will  have  an  alti-  ^ 

tude  of  50°.   Suppose  the  date  o^'y^''^^     ^^\ 

on   which  the  observation  is         /vS'^'^'^         ^\^ 
made  is  June  21  and  the  time    so-f  h^    '■,  \ 

of  day  is  8  p.m.  Suppose  the  /  '*a--\ —  -vf-  ---,.,  V"" 
right  ascension  of  the  star  in  J(i        \     ol  '\  \n 

question  is  approximately  16     VpJg...^  \/  ■     ''^_^--^ 
hours  and  that  its  declination      M  ^\         \        / 

is  —  16°.     The  problem  is  to       \  \^    \   / 

find  its  apparent  altitude  and  n^^^  J^' 

azimuth.  — ■^r-"""'^ 

The   steps    of    the    solution    Fia.  52. —Finding  the  altitude  and 

will  be  made  in  their  natural  ^'™"*^- 

order.  (1)  Since  June  21  is  three  months  after  March  21, 
the  right  ascension  of  the  sun  on  that  date  is  6  hours. 
(2)  Since  the  time  of  day  is  8  p.m.,  and  the  right  ascension  is 
coimted  eastward,  the  right  ascension  of  the  meridian  is 
6  +  8  =14  hours.  (3)  Since  the  right  ascension  of  the  star 
is  16  hours,  its  hour  angle  is  2  hours  east,  and  it  is  on  the 
hour  circle  PCP'.  (4)  Since  its  declination  is  —16°,  it  is 
16°  south  from  C  toward  P'  and  at  the  point  A.  Now  draw 
a  vertical  circle  from  Z  through  A,  cutting  the  horizon  at  B. 
The  altitude  is  BA,  which  is  aboi^it  22°.  The  azimuth  is 
SWNEB,  which  is  about  320°. 

71.  Finding  the  Right  Ascension  and  Declination  when 
the  Altitude  and  Azimuth  are  Given.  —  The  problem  of 
finding  the  right  ascension  and  declination  when  the  altitude 
and  azimuth  are  given  is  the  converse  of  that  treated  in 
Art.  69.     It  can  also  be  conveniently  solved  in  four  steps. 

In  the  first  step,  the  right  ascension  of  the  sun  is  obtained, 
and  in  the  second,  the  right  ascension  of  the  meridian  is 
found.    These  steps  are,  of  course,  the  same  as  those  given 


134       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  iv,  71 


in  Art.  69.  The  third  step  is  to  draw  through  the  position 
of  the  given  object  an  hour  circle  which,  from  its  defini- 
tion, reaches  from  one  pole  of  the  sky  to  the  other  and  cuts 
the  equator  at  right  angles.  The  fourth  step  is  to  estimate 
the  hour  angle  of  the  hour  circle  drawn  in  Step  3  and  the 
distance  of  the  star  from  the  equator  measured  along  the 
hour  circle.  Then  the  right  ascension  of  the  object  is  equal 
to  the  right  ascension  of  the  meridian  plus  the  hour  angle 
of  the-object  if  it  is  east,  and  minus  the  hour  angle  if  it  is 
west ;  and  the  declination  of  the  object  is  simply  its  distance 
from  the  equator. 

72.  Illustrative  Example  for  Finding  Right  Ascension 
and  Declination.  —  Suppose  the  date  of  the  observation  is 
May  6  and  that  the  time  of  day  is  8  p.m.  Suppose  the 
observer's  latitude  is  40°  north.  Suppose  he  sees  a  bright 
star  whose  altitude  is  estimated  to  be  35°  and  whose  azimuth 

is  estimated  to  be  60°.  Its 
right  ascension  and  decUnation 
are  required,  and  after  they 
have  been  obtained  it  can  be 
found  from  Table  I,  p.  144, 
what  star  is  observed. 

The  right  ascension  of  the 
sun  on  May  6  is  3  hours  and 
the  right  ascension  of  the  me- 
ridian at  8  P.M.  is  11  hours. 
The  star  then  is  at  the  point 
A,  Fig.  53,  where  BA  =35° 
and  SB  =  60°.  The  part  of 
the  vertical  circle  BA  is  much 
less  foreshortened  than  AZ  by  the  projection  of  the  celestial 
sphere  on  a  plane,  and  this  fact  must  be  remembered  in 
connection  with  the  drawing.  The  hour  circle  PAP'  cuts 
the  equator  at  the  point  C.  The  arc  QC  is  much  more  fore- 
shortened by  projection  than  CW.  Consequently,  it  is  seen 
that  the  hour  angle  of  the  star  is  3.5  hours  west.     Therefore 


Fig.  53.  —  Finding  the  right  asceii' 
sion  and  declination. 


CH.  IV,  73]     REFERENCE   POINTS  AND   LINES  135 

its  right  ascension  is  11-3.5  =  7.5  hours  approximately. 
It  is  also  seen  that  the  star  is  approximately  5°  north  of  the 
equator.  On  referring  to  Table  I,  it  is  found  that  this  star 
must  be  Procyon. 

All  problems  of  the  same  class  can  be  solved  in  a  similar 
manner.  But  reUance  should  not  be  placed  in  the  diagrams 
alone,  especially  because  of  the  distortion  to  which  certain 
of  the  lines  are  subject.  The  diagrams  should  be  supple- 
mented, if  not  replaced,  by  actually  pointing  out  on  the 
sky  the  various  points  and  lines  which  are  used.  A  little 
practice  with  this  method  will  enable  one  to  solve  either 
the  problem  of  finding  the  altitude  and  azimuth,  or  that 
of  obtaining  the  right  ascension  and  declination,  with  an 
error  not  exceeding  5°  or  10°. 

73.  Other  Problems  Connected  with  Position.  —  There 
are  two  other  problems  of  some  importance  which  naturally 
arise. ^  The  first  is  that  of  finding  the  time  of  the  year  at 
which  a  star  of  given  right  ascension  will  be  on  the  meridian 
at  a  time  in  the  evening  convenient  for  observation. 

In  order  to  make  the  problem  concrete,  suppose  the  time 
in  question  is  8  p.m.  The  right  ascension  of  the  sun  is  then 
8  hours  less  than  the  right  ascension  of  the  meridian.  Since 
the  object  is  supposed  to  be  on  the  meridian,  the  right  ascen- 
sion of  the  sun  wiU  be  8  hours  less  than  that  of  the  object. 
To  find  the  time  of  the  year  at  which  the  sun  has  a  given 
right  ascension,  it  is  only  necessary  to  count  forward  from 
March  21  two  hours  for  each  month.  For  example,  if  the 
object  is  Arcturus,  whose  right  ascension  is  14  hours,  the 
right  ascension  of  the  sun  is  14  —  8  =  6  hours,  and  the  date 
is  June  21. 

The  second  problem  is  that  of  finding  the  time  of  day 
at  which  an  object  whose  right  ascension  is  given  will  be  on 
the  meridian  or  horizon  on  a  given  date.  A  problem  of  this 
character  will  naturally  arise  in  connection  with  the 
announcement  of  the  discovery  of  a  comet  or  some  other 
object  whose  appearance  in  a  given  position  would  be  con- 


136       AN   INTRODUCTION   TO   ASTRONOMY     [ch.  iv,  73 

spicuous  only  for  a  short  time.  This  problem  is  solved  by- 
first  finding  the  right  ascension  of  the  sun  on  the  date,  and 
then  taking  the  difference  between  this  result  and  the  right 
ascension  of  the  object.  This  gives  the  hour  angle  of  the 
sun  at  the  required  time.  If  the  sun  is  west  of  the  meridian, 
its  hour  angle  is  the  time  of  day ;  if  it  is  east  of  the  meridian, 
its  hour  angle  is  the  number  of  hours  before  noon. 

VII.  QUESTIONS 

1.  Make  a  table  showing  the  correspondences  of  the  points, 
circles,  and  coordinates  of  the  horizon,  equator,  and  ecUptic  systems 
with  those  of  the  geographical  system. 

2.  What  are  the  altitude  and  azimuth  of  the  zenith,  the  east 
point,  the  north  pole?  What  are  the  angular  distances  from  the 
zenith  to  the  pole  and  to  the  point  where  the  equator  crosses  the 
meridian  in  terms  of  the  latitude  I  of  the  observer  ? 

3.  Estimate  the  horizon  coordinates  of  the  sun  at  10  o'clock  this 
morning ;  at  10  o'clock  this  evening. 

4.  Describe  the  complete  diurnal  motions  of  stars  near  the  pole. 
What  part  of  the  sky  for  an  observer  in  latitude  40°  is  always  above 
the  horizon  ?    Always  below  the  horizon  ? 

6.  How  long  is  required  for  the  sky  apparently  to  turn  1°? 
Through  what  angle  does  it  apparently  turn  in  1  minute  ? 

6.  Are  there  positions  on  the  earth  from  which  the  diurnal 
motions  of  the  stars  are  along  parallels  of  altitude  ?  Along  vertical 
circles  ? 

7.  Develop  a  rule  for  finding  the  hour  angle  of  the  vernal 
equinox  on  any  date  at  any  time  of  day. 

8.  Find  the  altitude  and  azimuth  of  the  vernal  equinox  at 
9  A.M.  to-day. 

—     9.  Given:  Rt.  asc.  =  19  hrs.,  declination  = +20°,  date  =  July 21, 
time  =  8  p.m.  ;  find  the  altitude  and  azimuth. 

10.  Find  the  altitude  and  the  azimuth  (constructing  a  diagram) 
of  each  of  the  stars  given  in  Table  I,  p.  144,  at  8  p.m.  to-day. 

11.  If  a  star  whose  right  ascension  is  18  hours  is  on  the  meridian 
at  8  P.M.,  what  is  the  date  ? 

12.  At  what  time  of  the  day  is  a  star  whose  right  ascension  is 
14  hours  on  the  meridian  on  May  21  ? 

13.  At  what  time  of  the  day  does  a  comet  whose  right  ascension 
is  4  hours  and  dechnation  is  zero  rise  on  Sept.  21  ? 

14.  The  Leonid  meteors  have  their  radiant  at  right  ascension 


CH.  IV,  73]     REFERENCE   POINTS  AND   LINES  137 

10  hours  and  they  appear  on  Nov.  14.    At  what  time  of  the  night 
are  they  visible  ? 

15.  What  is  the  right  ascension  of  the  point  on  the  celestial  sphere 
toward  which  the  earth  is  moving  on  June  21  ? 

16.  What  are  the  altitude  and  azimuth  of  the  point  toward  which 
the  earth  is  moving  to-day  at  noon  ?     At  6  p.m.  ?    At  midnight  ? 

At  6  A.M.  ? 

17.  Observe  some  conspicuous  star  (avoid  the  planets),  estimate 
its  altitude  and  azimuth,  approximately  determine  its  right  ascen- 
sion and  declination  (Art.  71),  and  with  these  data  identify  it  in 
Table  I,  p.  144. 


Fig.  54.  —  The  40-inch  telescope  of  the  Yerkes  Observatory. 
138      " 


CHAPTER  V 
THE   CONSTELLATIONS 

74.  Origin  of  the  Constellations.  —  A  moment's  obser- 
vation of  the  sky  on  a  clear  and  moonless  night  shows  that 
the  stars  are  not  scattered  uniformly  over  its  surface.  Every 
one  is  acquainted  with  such  groups  as  the  Big  Dipper  and 
the  Pleiades.  This  natural  grouping  of  the  stars  was' ob- 
served in  prehistoric  times  by  primitive  and  childlike  peoples 
who  ima^ned  the  stars  formed   outhnes  of  various  living 

'  creatures,  and  who  often  wove  about  them  the  most  fantastic 
romances. 

The  earliest  list  of  constellations,  still  in  existence,  is  that 
of  Ptolemy  (about  140  A.D.),  who  enumerated,  described, 
and  located  48  of  them.  These  constellations  not  only  did 
not  entirely  cover  the  part  of  the  sky  visible  from  Alexandria, 
where  Ptolemy  lived,  but  they  did  not  even  occupy  all  of 
the  northern  sky.  In  order  to  fill  the  gaps  and  to  cover  the 
southern  sky  many  other  constellations  were  added  from 
time  to  time,  though  some  of  them  have  now  been  aban- 
doned. The  lists  of  Argelander  (1799-1875)  in  the  northern 
heavens,  and  the  more  recent  ones  of  Gould  in  the  southern 
heavens,  contain  80  constellations,  and  these  are  the  ones 
now  generally  recognized. 

75.  Naming  the  Stars.  —  The  ancients  gave  proper 
names  to  many  of  the  stars,  and  identified  •  the  others  by 
describing  their  relations  to  the  anatomy  of  the  fictitious 
creatures  in  which  they  were  situated.  For  example,  there 
were  Sirius,  Altair,  Vega,  etc.,  with  proper  names,  and 
"  The  Star  at  the  End  of  the  Tail  of  the  Little  Bear  "(Po- 
laris), "  The  Star  in  the  Eye  of  the  Bull  "  (Aldebaran),  etc., 
designated  by  their  positions. 

139 


140      AN   INTRODUCTION  TO  ASTRONOMY      [ch.  v,  75 

In  modern  times  the  names  of  40  or  50  of  the  most  con- 
spicuous stars  are  frequently  used  by  astronomers  and 
writers  on  astronomy;  the  remainder  are  designated  by 
letters  and  numbers.  A  system  in  very  common  use,  that 
introduced  by  Bayer  in  1603,  is  to  give  to  the  stars  in  each 
constellation,  in  the  order  of  their  brightness,  the  names  of 
the  letters  of  the  Greek  alphabet  in  their  natural  order. 
In  connection  with  the  Greek  letters,  the  genitive  of  the  name 
of  the  constellation  is  used.  For  example,  the  brightest 
star  in  the  whole  sky  is  Sirius,  in  Canis  Major.  Its  name 
according  to  the  system  of  Bayer  is  Alpha  Canis  Majoris. 
The  second  brightest  star  in  Perseus,  whose  common  name 


^'^•., 

Z£TA-^ 

• 

EPS/LOf^-^ 

POLARIS 

DELTA    '^ 

GAMMA. 

^  ALPHA 
■^  BETA 

Fig.  55.  —  The  Big  Dipper  and  the  Pole  Star. 

is  Algol,  in  this  system  is  called  Beta  Persei.  After  the 
Greek  letters  are  exhausted  the  Roman  letters  are  used,  and 
then  follow  numbers  for  the  stars  in  the  order  of  their  bright- 
ness. While  this  is  the  general  rule,  there  are  numerous 
exceptions  in  naming  the  stars,  for  example,  in  the  case  of 
the  stars  which  constitute  the  Big  Dipper  (Fig.  55). 

About  1700,  Flamsteed  published  a  catalogue  of  stars  in 
which  he  numbered  those  in  each  constellation  according  to 
their  right  ascensions  regardless  of  their  brightness.  In 
modern  catalogues  the  stars  are  usually  given  in  the  order 
of  their  right  ascension  and  no  reference  is  made  either  to  the 
constellation  to  which  they  belong  or  to  their  apparent 
brightness. 


CH.  V,  76]  THE   CONSTELLATIONS  141 

76.  Star  Catalogues.  —  Star  catalogues  are  lists  of  stars, 
usually  all  above  a  given  brightness,  in  certain  parts  of  the 
sky,  together  with  their  right  ascensions  and  declinations  on 
a  given  date.  It  is  necessary  to  give  the  date,  for  the  stars 
slowly  move  with  respect  to  one  another,  and  the  reference 
points  and  hnes  to  which  their  positions  are  referred  are  not 
absolutely  fixed.  The  mos^  important  variation  in  the  posi- 
tion of^the  reference  points  and  lines  is  due  to  the  precession 
of  the  equinoxes  (Art.  47). 

The  earliest  known  star  catalogue  is  one  of  1080  stars  by 
Hipparchus  for  the  epoch  125  b.c.  Ptolemy  revised  it  and 
reduced  the  star  places  to  the  epoch  150  a.d.  Tycho  Brahe 
made  a  catalogue  of  1005  stars  in  1580,  about  30  years  be- 
fore the  invention  of  the  telescope.  Since  the  invention  of 
the  telescope  and  the  revival  of  science  in  Europe,  numerous 
catalogues  have  been  made,  containing  in  some  cases  more 
than  100,000  stars.  While  the  positions  in  all  these  cata- 
logues are  very  accurately  given,  compared  even  to  the 
work  of  Tycho  Brahe,  they  are  not  accurate  enough  for 
certain  of  the  most  refined  work  in  modern  times.  To  meet 
these  needs,  a  nimiber  of  catalogues,  containing  a  Umited 
number  of  stars  whose  positions  have  been  determined 
with  the  very  greatest  accuracy,  have  been  made.  The 
most  accurate  of  these  is  the  Prehminary  General  Catalogue 
of  Boss,  in  which  the  positions  of  6188  stars  are  given. 

A  project  for  photographing  the  whole  heavens  by  in- 
ternational cooperation  was  formulated  at  Paris  in  1887. 
The  plan  provided  that  each  plate  should  cover  4  square 
degrees  of  the  sky,  and  that  they  should  overlap  so  that  the 
whole  sky  would  be  photographed  twice.  The  nimiber  of 
plates  required,  therefore,  is  nearly  22,000.  On  every  plate 
a  number  of  stars  are  photographed  whose  positions  are 
already  known  from  direct  observations.  The  positions  of 
the  other  stars  on  the  plate  can  then  be  determined  by  meas- 
uring with  a  suitable  machine  their  distances  and  direc- 
tions from  the  known  stars.     This  work  can,  of  course,  be 


142       AN  INTRODUCTION  TO  ASTRONOMY       [ch.  v,  76 

carried  out  at  leisure  in  an  astronomical  laboratory.  On 
these  plates,  most  of  which  have  already  been  secured,  there 
will  be  shown  in  all  about  8,000,000  different  stars.  In  the 
first  catalogue  based  on  them  only  aboijt  1,300,000  of  the 
brightest  stars  will  be  given. 

The  photographic  catalogue  was  an  indirect  outgrowth 
of  photographs  of  the  great  comet  of  1882  taken  by  Gill 
at  the  Cape  of  Good  Hope.  The  number  of  star  images 
obtained  on  his  plates  at  once  showed  the  possibihties  of 
making  catalogues  of  stars  by  the  photographic  method. 
In  1889  he  secured  photographs  of  the  whole  southern  sky 
from  declination  —  19°  south,  and  the  enormous  labor  of 
measuring  the  positions  of  the  350,000  star  images  on  these 
plates  was  carried  out  by  Kapteyn,  of  Groningen,  Holland. 

77.  The  Magnitudes  of  the  Stars.  —  The  magnitude  of 
a  star  depends  upon  the  amount  of  light  received  from  it 
by  the  earth,  and  is  not  determined  altogether  by  the  amount 
of  light  it  radiates,  for  a  small  star  near  the  earth  liiight 
give  the  observer  more  light  than  a  much  larger  one  farther 
away.  It  is  clear  from  this  fact  that  the  magnitude  of  a 
star  depends  upon  its  actual  brightness  and  also  upon  its 
distance  from  the  observer. 

The  stars  which  are  visible  to  the  unaided  eye  are  divided 
arbitrarily  into  6  groups,  or  magnitudes,  depending  upon 
their  apparent  brightness.  The  20  brightest  stars  con- 
stitute the  first-magnitude  group,  and  the  faintest  stars 
which  can  be  seen  by  the  ordinary  eye  on  a  clear  night  are 
of  the  sixth  magnitude,  the  other  four  magnitudes  being  dis- 
tributed between  them  so  that  the  ratio  of  the  brightness 
of  one  group  to  that  of  the  next  is  the  same  for  all  consecu- 
tive magnitudes.  The  definition  of  what  shall  be  exactly  the 
first  magnitude  is  somewhat  arbitrary;  but  a  first-magni- 
tude star  has  been  taken  to  be  approximately  equal  to  the 
average  brightness  of  the  first  20  stars.  The  sixth-magni- 
tude stars  are  about  tott  as  bright  as  the  average  of  the  first 
group,  and,  in  order  to  make  the  ratio  from  one  magnitude 


CH.  V,  78]  THE   CONSTELLATIONS  143 

to  the  pther  perfectly  definite,  it  has  been  agreed  that  the 
technical  sixth-magnitude  stars  shall  be  those  which  are 
exactly  xiv  ^  bright  as  the  technical  first-magnitude  stars. 
The  problem  arises  of  finding  what  the  ratio  is  for  succes- 
sive magnitudes. 

Let  r  be  the  ratio  of  light  received  from  a  star  of  one 
magnitude  to  that  received  from  a  star  of  the  next  fainter 
magnitude.  Then  stars  of  the  fifth  magnitude  are  r  times 
brighter  than  those  of  the  sixth,  and  those  of  the  fourth  are 
r  times  brighter  than  those  of  the  fifth,  and  they  are  there- 
fore r^  times  brighter  than  those  of  the  sixth.  By  a  repeti- 
tion of  this  process  it  is  found  that  the  first-magnitude  stars 
are  r^  times  brighter  than  those  of  the  sixth  magnitude. 
Therefore  r*  =  100,  from,  which  it  is  found  that  r  =  2.512.  .  .  . 

Since  the  amount  of  light  received  from  different  stars 
varies  almost  continuously  from  the  faintest  to  the  brightest, 
it.  is  necessary  to  introduce  fractional  magnitudes.  For 
example,  if  a  star  is  brighter  than  the  second  magnitude  and 
fainter  than  the  first,  its  magnitude  is  between  1  and  2. 
A  step  of  one  tenth  of  a  magnitude  is  such  a  ratio  that, 
when  repeated  ten  times,  it  gives  the  value  2.512.  ...  It 
is  found  by  computation,  which  can  easily  be  carried  out  by 
logarithms,  that  a  first-magnitude  star  is  1.097  times  as 
bright  as  a  star  of  magnitude  1.1.  The  ratio  of  brightness 
of  a  star  of  magnitude  1.1  to  that  of  a  star  of  1.2  is  like- 
wise 1.097;  and,  consequently,  a  star  of  magnitude  1  is 
1.097  X  1.097  =  1.202  times  as  bright  as  a  star  of  magni- 
tude 1.2. 

A  star  which  is  2.512  times  as  bright  as  a  first-magnitude 
star  is  of  magnitude  0,  and  still  brighter  stars  have  negative 
magnitudes.  For  example,  Sirius,  the  brightest  star  in  the 
sky,  has  a  magnitude  of  —1.58,  and  the  magnitude  of  the 
full  moon  on  the  same  system  is  about  —12,  while  that  of 
the  sun  is  —26.7. 

78.  The  First-magnitude  Stars.  —  As  first-magnitude 
stars  are  conspicuous  and  relatively  rare  objects,  they  serve 


144       AN   INTRODUCTION   TO   ASTRONOMY       [ch.  v,  78 

as  guideposts  in  the  study  of  the  constellations.  All  of 
those  which  are  visible  in  the  latitude  of  the  observer  should 
be  identified  and  learned.  They  will,  of  course,  be  recog- 
nized partly  by  their  relations  to  neighboring  stars. 

In  Table  I  the  first  column  contains  the  names  of  the  first- 
magnitude  stars;  the  second,  the  constellations  in  which 
they  are  found ;  the  third,  their  magnitudes  according  to  the 
Harvard  determination ;  the  fourth,  their  right  ascensions ; 
the  fifth,  their  declinations ;  the  sixth,  the  dates  on  which 
they  cross  the  meridian  at  8  p.m.  ;  and  the  seventh,  the 
velocity  toward  or  from  the  earth  in  miles  per  second,  the 
negative  sign  indicating  approach  and  the  positive,  recession. 
Their  apparent  positions  at  any  time  can  be  determined 
from  their  right  ascensions  and  declinations  by  the  principles 
explained  in  Art.  69. 

Table  I 


Name 

CONSTELLAT 

Mag- 
ion       Nl- 

Right  As- 
cension 

Decli- 
nation 

On  Me- 
ridian 

Radial 
v^elocitt 

TUDE 

AT  8  P.M. 

Sinus   .     .     . 

Canis  Ma; 

or     -1.6 

6h41m 

-16°  36' 

Feb.  28 

-   5.6 

Canopus 

Carina 

.    -0.9 

6 

22 

-52    39 

Feb.  23 

-1-12.7 

Alpha 

Centauri    . 

Centaurus 

0.1 

14 

34 

-60    29 

June  29 

-13.8 

Vega 

Lyra 

0.1 

18 

34 

-1-38    42 

Aug.  30 

-  8.5 

Capella     .     . 

Auriga 

0.2 

5 

10  . 

-1-45   55 

Feb.,    5 

-1-19.7 

Arcturus  .     . 

Bootes 

.       0.2 

14 

12 

-1-19   37 

June  24 

-  2.4 

Rigel    .     .     . 

Orion   . 

0.3 

5 

11  . 

-  8   17 

Feb.    5 

-1-13.6 

Procyon    .     . 

Canis  Min 

or         0.5 

7 

35 

-1-  5   26 

Mar.  14 

-   2.5 

Achernar .     . 

Eridan\is 

0.6 

1 

35 

-57   40 

Dec.  16 

-1-10.0 

Beta 

Centauri    . 

Centaurus 

0.9 

13 

58 

-59   58 

June  21 

? 

Betelgeuze     . 

Orion   . 

0.9 

5 

51 

-f  7   24 

Feb.  15 

+13.0 

Altair  .     .     . 

Aquila 

0.9 

19 

47 

-1-  8   39 

Sept.  19 

-20.5 

Alpha  Cnicis 

Crux     . 

1.1 

12 

22 

-62   38 

May  29 

+  4.3 

Aldebaran     . 

Taurus 

1.1 

4 

31 

-fl6   21 

Jan.  26 

-1-34.2 

PoUux .     .     . 

Gemini 

1.2 

7 

40 

-1-28    14 

Mar.  15 

+  2.4 

Spica    .     .     . 

Virgo 

1.2 

13 

21 

-10   44 

June  12 

+   1.2  ( 

Antares     .     . 

Scorpius 

.        1.2 

16 

24 

-26    15 

July  27 

-   1.9 

Fomalhaut    . 

Piscis 

Australis 

1.3 

22 

53 

-30     4 

Nov.   8 

-t-  4.2 

Deneb .     .     . 

Cygnus 

.     .        1.3 

20 

39 

-1-44   59 

Oct.     4 

-  2.5 

Regulus    .     . 

Leo 

.     .        1.3 

10 

4 

-1-12   23 

Apr.  23 

-  5.0 

CH.  V,  80] 


THE   CONSTELLATIONS 


145 


79.   Number  of  Stars  in  the  First  Six  Magnitudes.  —  The 

i;iumber  of  stars  in  each  of  the  first  six  magnitudes  is  given 
in  Table  II.    The  sum  of  the  numbers  is  5000. 


Table  II 


First  Magnitude  ...  20 
Second  Magnitude  ...  65 
Third  Magnitude    .  .  190 


Fourth  Magnitude.  .  .  425, 
Fifth  Magnitude  .  .  .1100 
Sixth  Magnitude    .     .     .  3^00 


There  are,  therefore,  in  the  whole  sky  only  about  5000  stars 
which  are  visible  to  the  imaided  eye.  At  any  one  time 
only  half  the  sky  is  above  the  horizoh,  and  those  stars  which 
are  near  the  horizon  are  largely  extinguished  by  the  absorp- 
tion of  Ught  by  the  earth's  ,  atmosphere.  Therefore  one 
never  sees  at  one  time  more  than  about  2000  stars,  although 
the  general  impression  is  that  they  are  countless. 

It  is  seen  from  the  Table  II  that  the  number  of  stars  in 
each  magnitude  is  about  three  times  as  great  as  the  number 
in  the  preceding  magnitude.  This  ratio  holds  approxi- 
mately down  to  the  ninth  magnitude,  and  in  the  first  nine 
magnitudes  there  are  in  all  nearly  200,000  stars.  Since 
a  telescope  3  inches  in  aperture  will  show  objects  as  faint  as 
the  ninth  magnitude,  it  is  seen  what  enormous  aid  is  ob- 
tained from  optical  instruments.  Only  a  rough  guess  can 
be  made  respecting  the  number  of  stars  which  are  still 
fainter,  but  there  are  probably  more  than  300,000,000  of 
them  within  the  range  of  present  visual  and  photographic- 
instruments. 

80.  The  Motions  of  the  Stars.  —  The  stars  have  motions 
with  respect  to  one  another  which,  in  the  course  of  immense 
ages;  appreciably  change  the  outlines  of  the  constellations, 
but  which  have  not  made  important  alterations  in  the  visible 
sky  during  historic  times.  Nevertheless,  they  are  so  large 
that  they  must  be  taken  into  account  when  using  star  cata- 
logues in  work  of  precision. 


146        AN   INTRODUCTION   TO  ASTRONOMY     [ch.  v,  80 

One  result  of  the  motions  of  the  stars  is  that  they  drift 
with  respect  to  fixed  reference  points  and  lines.  The  yearly- 
change  in  the  position  of  a  star  with  respect  to  fixed  reference 
points  and  Unes  is  called  its  proper  motion.  The  largest 
known  proper  motion  is  that  of  an  eighth-magnitude  star 
in  the  southern  heavens,  whose  annual  displacement  on  the 
sky  is  about  8.7  seconds  of  arc.  The  shght  extent  to  which 
the  proper  motions  of  the  stars  can  change  the  appearance 
of  the  constellations  is  shown  by  the  fact  that  even  this 
star,  whose  proper  motion  is  more  than  100  times  the  average 
proper  motion  of  the  brighter  stars,  will  not  move  over  an 
apparent  distance  as  great  as  the  diameter  of  the  moon  in 
less  than  220  years. 

Another  component  of  the  motion  of  a  star  is  that  which  is 
in  the  line  joining  it  with  the  earth.  This  component  can 
be  measured  by  the  spectroscope  (Art.  222),  and  is  foimd 
to  range  all  the  way  from  a  velocity  of  approach  of  40  miles 
per  second  to  one  of  recession  with  the  same  speed ;  and 
in  some  cases  even  higher  velocities  are  encountered.  In 
the  course  of  immense  time  the  changes  in  the  distances  of 
the  stars  will  alter  their  magnitudes  appreciably;  but  the 
distances  of  the  stars  are  so  great  that  there  is  probably  no 
case  in  which  the  motion  of  a  star  toward  or  from  the  earth 
will  sensibly  change  its  magnitude  in  20,000  years. 

81.  The  Milky  Way,  or  Galaxy.  — The  Milky  Way  is  a 
hazy  band  of  hght  giving  indications  to  the  unaided  eye  of 
being  made  up  of  faint  stars ;  it  is  on  the  average  about  20° 
in  width  and  stretches  in  nearly  a  great  circle  entirely  around 
the  sky.  The  telescope  shows  that  it  is  made  up  of  milhons 
of  small  stars  which  can  be  distinguished  separately  only 
with  optical  aid.  It  is  clear  that  because  of  its  irregular 
form  and  great  width  its  position  cannot  be  precisely  de- 
scribed, but  in  a  general  way  its  location  is  defined  by  the 
fact  that  it  intersects  the  celestial  equator  at  two  places 
whose  right  ascensions  are  approximately  6  hours  40  minutes 
and  18  hours  40  minutes,  and  it  has  an  incUnation  to  the 


CH.  V,  82]  THE   CONSTELLATIONS  147 

equator  of  about  62°.  Or,  in  other  terms,  the  north  pole 
of  the  Milky  Way  is  at  right  ascension  about  12  hours  40 
minutes  and  at  decUnation  about  +  28°.  For  a  long  distance 
it  is  divided  more  or  less  qompletely  into  two  parts,  and  at 
one  place  in  the  southern  heavens  it  is  cut  entirely  across  by 
a  dark  streak.  A  very  interesting  feature  for  observers  in 
northern  latitudes  is  a  singular  dark  region  north  of  the  star 
Deneb. 

82.  The  Constellations  and  Their  Positions.  —  The  work 
on  reference  points  and  lines  in  the  preceding  chapter  to- 
gether with  the  discussions  so  far  given  in  this  chapter  are 
sufficient  to  prepare  for  the  study  of  the  constellations  with 
interest  and  profit,  and  the  student  should  not  stop  short 
of  an  actual  acquaintance  with  all  the  first-magnitude  stars 
and  the  principal  constellations  that  are  visible  in  his  latitude. 
Table  III  contains  a  list  of  the  constellations  and  gives  their 
positions.  The  numbers  at  the  top  show  the  degrees  of  dec- 
lination between  which  the  constellations  he,  the  numerals 
at  the  left '  show  their  right  ascensions,  and  the  numbers 
placed  in  connection  with  the  names  of  the  constellations 
give  the  nmnber  of  stars  in  them  which  are  easily  visible  to 
the  unaided  eye.  The  constellations  which  lie  on  the  ecliptic, 
or  the  so-called  zodiacal  constellations,  are  printed  in  italics. 

The  following  maps  show  the  constellations  from  the  north 
pole  to  —50°  declination.  When  Map  I  is  held  up  toward 
the  sky,  facing  north,  with  its  center  in  the  line  joining  the 
eye  with  the  north  pole,  and  with  the  hour  circle  having  the 
right  ascension  of  the  meridian  placed  directly  above 
its -center,  it  shows  the  circumpolar  constellations  in  their 
true  relations  to  one  another  and  to  the  horizon  and  pole. 
The  other  maps  are  to  be  used,  facing  south,  with  their  cen- 
ters held  on  a  line  joining  the  eye  to  the  celestial  equator, 
and  with  the  hour  circle  having  the  right  ascension  of  the 
meridian  held  in  the  plane  of  the  eye  and  the  meridian. 
When  they  are  placed  in  this  way,  they  show  the  constellations 
to  the  south  of  the  observer  in  their  true  relationships.     In 


148         AN   INTRODtJCTION   TO  ASTRONOMY  '  [ch.  v,  82 


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CH.  V,  83] 


THE   CONSTELLATIONS 


149 


order  to  apply  the  maps  according  to  these  instructions,  it 
is  necessary  to  know  the  right  ascension  of  the  meridian  for 
the  day  and  hour  in  question,  and.  it  can  be  computed  with 
sufficient  approximation  by  the  method  of  Art.  69. 

83.  Finding  the  Pole  Star.  —  The  first  step  to  be  taken 
in  finding  the  constellations,  either  from  their  right  ascen- 
sions and  decUnations  or  from  star  maps,  is  to  determine 
the  north-and-south  line.  It  is  defined  closely  enough  for 
present  purposes  by  the  position  of  the  pole  star. 

The  Big  Dipper  is  the  best  known  and  one  of  the  most 
conspicuous  groups  of  stars  in  the  northern  heavens.  It  is 
always  above  the 
horizon  for  an  ob- 
server in  latitude 
40°  north,  and,  be- 
cause of  its  defi- 
nite shape,  it  can 
never  be  mistaken 
for  any  other  group 
of  stars.  It  is 
made  up  of  7  stars 
of  the  second  mag- 
nitude which  form  the  outline  of  a  great  dipper  in  the  sky. 
Figure  56  is  a  photograph  of  this  group  of  stars  distinctly 
showing  the  dipper.  The  stars  Alpha  and  Beta  are  called 
The  Pointers  because  they  are  almost  directly  in  a  line  with 
the  pole  star  Polaris.  In  order  to  find  the  pole  star,  start  with 
Beta,  Fig.  55,  go  through  Alpha,  and  continue  abou^t  five 
times  the  distance  from  Beta  to  Alpha.  At  the  point  reached 
there  will  be  found  the  second-magnitude  star  Polaris  with 
no  other  one  so  bright  anywhere  in  the  neighborhood. 

Besides  defining  the  north-and-south  hne  and  serving  as 
a  guide  for  a  study  of  the  constellations  in  the  northern 
heavens,  the  pole  star  is  an  interesting  object  in  several 
other  respects.  It  has  a  faint  companion  of  the  ninth  mag- 
nitude, distant  from  it  about  18.5  seconds  of  arc.    This 


Fig.  56.  —  The  Big  Dipper. 


150         AN   INTRODUCTION   TO  ASTRONOMY     [ch.  v,  83 

faint  companion  cannot  be  seen  with  the  unaided  eye  be- 
cause, in  order  that  two  stars  may  be  seen  as  separate  objects 
without  a  telescope,  they  must  be  distant  from  each  other 
at  least  3  minutes  of  arc,  and,  besides,  they  must  not  be  too 
bright  or  too  faint.  The  brighter  of  the  two  components  of 
Polaris  is  also  a  double  star,  a  fact  which  was  discovered  by 
means  of  the  spectroscope  in  1899.  Indeed,  it  has  turned 
out  on  more  recent  study  at  the  Lick  Observatory  that  the 
principal  star  of  this  system  is  really  a  triple  sun. 

84.  Units  for  Estimating  Angular  Distances.  —  The  dis- 
tances between  stars,  as  seen  projected  on  the  celestial 
sphere,  are  always  given  in  degrees.  There  is,  in  fact,  no 
definite  content  to  the  statement  that  two  stars  seem  to  be 
a  yard  apart.  In  order  to  estimate  angular  distances,  it  is 
important  to  have  a  few  units  of  known  length  which  can 
always  be  seen. 

It  is  90°  from  the  horizon  to  the  zenith,  and  one  would 
suppose  that  it  would  be  a  simple  matter  to  estimate  half 
of  this  distance.  As  a  matter  of  fact,  few  people  place  the 
zenith  high  enough.  In  order  to  test  the  accuracy  with 
which  one  locates  it,  he  should  face  the  north  and  fix  his 
attention  on  the  star  which  he  judges  to  be  at  the  zenith, 
and  then,  keeping  it  in  view,,  turn  slowly  around  until  he 
faces  the  south.     The  first  trial  is  apt  to  furnish  a  surprise. 

The  altitude  of  the  pole  star  is  equal  to  the  latitude  of 
the  observer  which,  in  the  United  States,  is  from  25°  to  50°. 
This  unit  is  not  so  satisfactory  as  some  others  because  it 
depends  upon  the  position  of  the  observer  and  also  because 
it  is  more  difficult  to  estimate  from  the  horizon  to  a  star 
than  it  is  between  two  stars.  Another  large  unit  which  can 
always  be  observed  from  northern  latitudes  is  the  distance 
between  Alpha  Ursae  Majoris  and  Polaris,  which  is  28°. 
For  a  smaller  unit  the  distance  between  The  Pointers  in  the 
Big  Dipper,  which  is  5°  20',  is  convenient. 

85.  Ursa  Major  (The  Greater  Bear).  —  The  Big  Dipper, 
to  which  reference  has  already  been  made,  and  which  is  one 


CH.  V,  85]  THE   CON-STELLATIONS  151 

of  the  most  conspicuous  configurations  in-  the  northern 
heavens,  is  in  the  eastern  part '  of  the  constellation  Ursa 
Major  and  serves  to  locate  the  position  of  this  constellation. 
The  outline  of  the  Bear  extends  north,  south,  and  west  of 
the  bowl  of  the  Dipper  for  more  than  10° ;  but  all  the  stars 
in  this  part  of  the  sky  are  of  the  third  magnitude  or  fainter. 

According  to  the  Greek  legend,  Zeus  changed  the  nymph 
Callisto  into  a  bear  in  order  to  protect  her  from  the  jealousy 
of  his  wife  Hera.  While  the  transformed  Callisto  was  wan- 
dering in  the  forest,  she  met  her  son  Areas,  who  was  about  to 
slay  her  when  Zeus  intervened  and  saved  her  by  placing  them 
both  among  the  stars,  where  they  became  the  Greater  and 
the  Smaller  Bears.  Hera  was  still  unsatisfied  and  prevailed 
on  Oceanus  and  Thetis  to  cause  them  to  pursue  forever  their 
courses  around  the  pole  without  resting  beneath  the  ocean 
waves.  Thus  was  explained  the  circumpolar  motions  of 
those  stars  which  are  always  above  the  horizon. 

The  Pawnee  Indians-call  the  stars  of  the  bowl  of  the  Dip- 
per a  stretcher  on  which  a  sick  man  is  being  carried,  and  the 
first  one  in  the  handle  is  the  medicine  man. 

The  star  at  the  bend  of  the  handle  of  the  Dipper,  called 
Mizar  by  the  Arabs,  has  a  faint  one/uear  it  which  is  known 
as  Alcor.  Mizar  is  of  the  second  magnitude,  and  Alcor  is  of 
the  fifth.  Any  one  with  reasonably  good  eyes  can  see  the  two 
stars  as  distinct  objects,  without  optical  aid.  It  is  probable 
that  this  was  the  first  double  star  that  was  discovered.  The 
distance  of  11 '.5  between  them  is  so  great,  astronomically 
speaking,  that  it  is  no  longer  regarded  as  a  true  double  star. 
It  has  been  supposed  by  some  writers  that  the  word  Alcor 
is  derived  from  an  Arabic  word  meaning  the  test,  and  the 

'  East  and  west  on  the  sky  must  be  understood  to  be  measured  along 
declination  circles.  Consequently,  near  the  pole  east  may  have  any  direc- 
tion with  respect  to  the  horizon.  Above  the  pole,  east  on  the  sky  is  toward 
the  eastern  part  of  the  horizon,  while  below  the  pole  it  is  toward  the  western 
part  of  the  horizon.  All  statements  of  direction  in  descriptions  of  the 
constellations  refer  to  directions  on  the  sky  unless  otherwise  indicated,  and 
care  inust  be  taken  not  to  understand  them  in  any  other  sense. 


152        AN   INTRODUCTION  TO  ASTRONOMY    [ch.  v,  85 

Arabs  are  said  to  have  tested  their  eyesight  on  it.  The 
Pawnee  Indians  call  it  the  Medicine  Man's  Wife's  Dog. 

The  star  Mizar  itself  is  a  fine  telescopic  double,  the  first 
one  ever  discovered ;  the  two  components  are-  distant  from 
each  other  14  ".6  and  can  be  seen  separately  with  a  3-inch 
telescope.  The  distance  from  the  earth  to  Mizar,  according 
to  the  work  of  Ludendorff,  is  4,800,000  times  as  far  as  from 
the  earth  to  the  sun,  and  about  75  years  are  required  for 
light  to  come  from  it  to  us.  The  star  appears  to  be  faint 
only  because  of  its  immense  distance,  for,  as  a  matter  of 
fact,  it  radiates  115  times  as  much  light  as  is  given  out  by 
the  sun.  The  actual  distance  even  from  Mizar  to  Alcor, 
which  is  barely  discernible  with  the  unaided  eye,  is  16,000 
times  as  far  as  from  the  earth  to  the  sun. 

The  first  of  a  series  of  very  important  discoveries  was  made 
by  E.  C.  Pickering,  in  1889,  by  spectroscopic  observations 
of  the  brighter  component  of  Mizar.  It  was  foimd  by 
methods  which  will  be  discussed  in  Arts.  285  and  286  that 
this  star  .is  itself  a  double  in  which  the  components  are  so 
close  together  that  they  cannot  be  distinguished  separately 
with  the  aid  of  any  existing  telescope.  Such  a  star  is  called 
a  spectroscopic  binary.  The  complete  discussion  showed 
that  the  brighter  component  of  Mizar  is  composed  of  two 
great  suns  whose  combined  mass  is  many  times  that  of- our 
sun,  and  that  they  revolve  about  their  common  center  of 
gravity  at  a  distance  of  25,000,000  miles  from  each  other  in 
a  period  of  20.5  days. 

86.  Cassiopeia  (The  Woman  in  the  Chair).  — To  find 
Cassiopeia  go  from  the  middle  of  the  handle  of  the  Big  Dipper 
through  Polaris  and  about  30°  beyond.  The  constellation 
will  be  recognized  because  the  principal  stars  of  which  it 
is  composed,  ranging  in  magnitude  from  the  second  to  the 
fourth,  form  a  zigzag,  or  letter  W.  When  it  is  tilted  in  a 
particular  way  as  it  moves  around  the  pole  in  its  diurnal 
motion,  it  has  some  resemblance  to  the  outline  of  a  chair. 
The  brightest  of  the  7  stars  in  the  W  is  the  one  at  the  bottom 


CH.  V,  88]  THE   CONSTELLATIONS  153 

of  its  second  part,  and  a  2-inch  telescope  will  show  that  it 
is  a  double  star  whose  colors  are  described  as  rose  and  blue. 

One  of  the  most  interesting  objects  in  this  constellation  is 
the  star  Eta  Cassiopeise,  which  is  near  the  middle  of  the  third 
stroke  of  the  W  and  about  2°  from  Alpha.  It  is  a  fine 
double  which  can  be  separated  with  a  3-inch  telescope. 
The  two  stars  are  not  only  apparently  close  together,  but 
actually  form  a  physical  system,  revolving  around  their 
common  center  of  gravity  in  a  period  of  about  200  years. 
If  there  are  planets  revolving  around  either  of  these  stars, 
their  phenomena  of  night  and  day  and  their  seasons  must 
be  very  comphcated. 

In  1572  a  new  star  suddenly  blazed  forth  in  Cassiopeia 
and  became  brighter  than  any  other  one  in  the  sky.  It 
caught  the  attention  of  Tycho  Brahe,  who  was  then  a  young 
man,  and  did  much  to  stimulate  his  interest  in  astronomy. 

87.  How  to  Locate  the  Equinoxes.  —  It  is  advantageous 
to  know  how  to  locate  the  equinoxes  when  the  positions  of 
objects  are  defined  by  their  right  ascensions  and  declinations. 
To  find  the  vernal  equinox,  draw  a  line  from  Polaris  through 
the  most  westerly  star  in  the  W  of  Cassiopeia,  and  continue 
it  90°.  The  point  where  it  crosses  the  equator  is  the  vernal 
equinox  which,  unfortunately,  has  no  bright  stars  in  its 
neighborhood. 

If  the  vernal  equinox  is  below  the  horizon,  the  autumnal 
equinox  may  be  conveniently  used.  One  or  the  other  of 
them  is,  of  course,  always  above  the  horizon.  To  find  the 
autumnal  equinox,  draw  a  line  from  Polaris  through  Delta 
Ursse  Majoris,  or  the  star  where  the  handle  of  the  Big 
Dipper  joins  the  dipper,  and  continue  it  90°  to  the  equator. 
The  autumnal  equinox  is  in  Virgo.  This  constellation 
contains  the  first-magnitude  star  Spica,  which  is  about  10° 
south  and  20°  east  of  the  autumnal  equinox. 

88.  Lyra  (The  Lyre,  or  Harp).  —  Lyra  is  a  small  but 
very  interesting  constellation  whose  right  ascension  is  about 
18.7  hours  and  whose  declination  is  about  40°  north.    It  is. 


154         AN    INTRODUCTION   TO   ASTRONOMY     [ch.  v,  88 

therefore,  about  50°  from  the  pole,  and  its  position  can  easily 
be  determined  by  using  the  directions  for  finding  the  vernal 
and  autumnal  equinoxes.  Or,  its  distance  east  or  west  of 
the  meridian  can  be  determined  by  the  methods  of  Art. 
69.  With  an  approximate  idea  of  its  location,  it  can  always 
be  found  because  it  contains  the  brilhant  Huish-white, 
first-magnitude  star  Vega.  If  there  should  be  any  doubt  in 
regard  to  the  identification  of  Vega,  it  can  always  be  dis- 
pelled by  the  fact  that  this  star,  together  with  two  fourth- 
magnitude  stars,  Epsilon  and  Zeta  Lyrse,  form  an  equi- 
lateral triangle  whose  sides  are  about  2°  in  length.  There 
are  no  other  stars  so  near  Vega,  and  there  is  no  other  con- 
figuration of  this  character  in  the  whole  heavens. 

As  was  stated  in  Art.  47,  the  attractions  of  the  moon  and 
sun  for  the  equatorial  bulge  of  the  earth  cause  a  precession 
of  the  earth's  equator,  and  therefore  a  change  in  the  location 
of  the  pole  of  the  sky.  About  12,000  years  from  now  the 
north  pole  will  be  very  close  to  Vega.  What  a  splendid 
pole  star  it  will  make  !  It  is  approaching  us  at  the  rate  of 
8.5  miles  per  second,  but  its  distance  is  so  enormous  that 
even  this  high  velocity  will  make  no  appreciable  change  in 
its  brightness  in  the  next  12,000  years.  The  distance  of 
Vega  is  not  very  accurately  known,  but  it  is  probably  more 
than  8,000,000  times  as  far  from  the  earth  as  the  earth  is 
from  the  sun.  At  its  enormous  distance  the  sun  would  ap- 
pear without  a  telescope  as  a  faint  star  nearly  at  the  limits 
of  visibility.  Another  point  of  interest  is  that  the  sun  with 
all  its  planets  is  moving  nearly  in  the  direction  of  Vega  at 
the  rate  of  about  400,000,000  miles  a  year. 

The  star  Epsilon  Lyrse,  which  is  about  2°  northeast  of 
Vega,  is  an  object  which  should  be  carefully  observed.  It  is 
a  double  star  in  which  the  apparent  distance  between  the 
two  components  is  207".  They  are  barely  distinguishable 
as  separate  objects  with  the  unaided  eye  even  by  persons 
of  perfect  eyesight.  It  is  a  noteworthy  fact  that,  so  far  as 
is  known,  this  star  was  not  seen  to  be  a  double  by  the  Arabs, 


CH.  V,  88]  THE   CONSTELLATIONS  155 

the  early  Greeks,  or  any  primitive  peoples.  A  century  ago 
astronomers  gave  their  ability  to  separate  this  pair  without 
the  use  of  the  telescope  as  proof  of  their  having  exceptionally 
keen  sight.  Perhaps  with  the  more  exacting  use  to  which 
the  eyes  of  the  human  race  are  being  subjected,  they  are 
actually  improving  instead  of  deteriorating  as  is  commonly 
supposed. 

Although  the  angular  distance  between  the  two  compor 
nents  of  Epsilon  Lyrae  seems  small,  astronomers  regularly 
measure  one  two-thousandth  of  this  angle.  The  discovery 
of  Neptune  was  based  on  the  fact  that  in  60  years  it  had 
pulled  Uranus  from  its  predicted  place,  as  seen  from  the 
earth,  only  a  little  more  than  half  of  the  angular  distance 
between  the  components  of  this  double  star.  When  Epsilon 
Lyrse  is  viewed  through  a  telescope  of  5  or  6  inches'  aperture, 
it  presents  a  great  surprise.  The  two  components  are  found 
to  be  so  far  apart  in  the  telescope  that  they  can  hardly  be 
seen  at  the  same  time,  and  a  little  close  attention  shows  that 
each  of  them  also  is  a  double.  That  is,  the  faint  object 
Epsilon  Lyrae  is  a  magnificent  system  of  four  suns. 

About  5°.5  south  of  Vega  and  3°  east  is  the  third-magni- 
tude star  Beta  Lyrse.  It  is  a  very  remarkable  variable 
whose  brightness  changes  by  nearly  a  magnitude  in  a  period 
of  12  days  and  22  hours.  The  variability  of  this  star  is  due 
to  the  fact  that  it  is  a  double  whose  plane  of  motion  passes 
nearly  through  the  earth  so  that  twice  in  each  complete 
revolution  one  star  eclipses  the  other.  A  detailed  study  of 
the  way  in  which  the  hght  of  this  star  varies  shows  that  the 
components  are  stars  whose  average  density  is  approximately 
that  of  the  earth's  atmosphere  at  sea  level. 

About  2°.5  southeast  of  Beta  Lyrse  is  the  third-magnitude 
star  Gamma  Lyrse.  On  a  line  joining  these  two  stars  and 
about  one  third  of  the  distance  from  Beta  is  a  ring,  or  an- 
nular, nebula,  the  only  one  of  the  few  that  are  known  that 
can  be  seen  with  a  small  telescope.  It  takes  a  large  telescope, 
however,  to  show  much  of  its  detail. 


156         AN   INTRODUCTION   TO   ASTRONOMY     [ch.  v,  89 

89.  Hercules  (The  Kneeling  Hero).  —  Hercules  is  a  very- 
large  constellation  lying  west  and  southwest  of  Lyra.  It 
contains  no  stars  brighter  than  the  third  magnitude,  but  it 
can  be  recognized  from  a  trapezoidal  figure  of  5  stars  which 
are  about  20°  west  of  Vega.  The  base  of  the  trapezoid,  which 
is  turned  to  the  north  and  slightly  to  the  east,  is  about  6° 
long  and  contains  two  stars  in  the  northeast  corner  which 
are  of  the  third  and  fourth  magnitudes.  The  star  in  the 
southeast  corner  is  of  the  fourth  magnitude,  and  the  others 
are  of  the  third  magnitude.  On  the  west  side  of  the  trape- 
zoid, about  one  third  of  the  distance  from  the  north  end,  is 
one  of  the  finest  star  clusters  in  the  whole  heavens,  known  as 
Messier  13.  It  is  barely  visible  to  the  unaided  eye  on  a 
clear  dark  night,  appearing  as  a  little  hazy  star ;  but  through 
a  good  telescope  it  is  seen  to  be  a  wonderful  object,  containing 
more  than  5000  stars  (Fig.  171)  which  are  probably  com- 
parable to  our  own  sun  in  dimensions  and  brilliancy.  The 
cluster  was  discovered  by  Halley  (1656-1742),  but  derives 
its  present  name  from  the  French  comet  hunter  Messier 
(1730-1817),  who  did  all  of  his  work  with  an  instrument  of 
only  2.5  inches'  aperture. 

90.  Scorpius  (The  Scorpion).  —  There  are  12  constella- 
tions, one  for  each  month,  which  lie  along  the  ecliptic  and 
constitute  the  zodiac.  Scorpius  is  the  ninth  of  these  and  the 
most  brilliant  one  of  all.  In  fact,  it  is  one  of  the  finest  group 
of  stars  that  can  be  seen  from  our  latitude.  It  is  60°  straight 
south  of  Hercules  and  can  always  be  easily  recognized  by 
its  fiery  red  first-magnitude  star  Antares,  which,  in  light- 
giving  power,  is  equal  to  at  least  200  suns  such  as  ours. 
The  word  Antares  means  opposed  to,  or  rivaUng,  Mars, 
the  ffed  planet  associated  with  the  god  of  war.  Antares  is 
represented  as  occupying  the  position  of  the  heart  of  a  scor- 
pion. About  7"  west  of  Antares  is  a  faint  green  star  of  the 
sixth  magnitude  which  can  be  seen  through  a  5-  or  6-inch 
telescope  under  good  atmospheric  conditions.  About  5° 
northwest  of  Antares  is  a  very  compact  and  fine  cluster. 


CH.  V,  93] 


THE   CONSTELLATIONS 


157 


Messier  80.    Scorpius  lies  in  one  of  the  richest  and  most 
varied  parts  of  the  Milky  Way. 

According  to  the  Greek  legend,  Scorpius  is  the  monster 
that  killed  Orion  and  frightened  the  horses  of  the  sun  so  that 
Phaeton  was  thrown  from  his  chariot  when  he  attempted  to 
drive  them. 

91.  Corona,  Borealis  (The  Northern  Crown),  — Just  west 
of  the  great  Hercules  lies  the  httle  constellation  Corona 
Borealis.  It  is  easily  recognized  by  the  semicircle,  or  crown, 
of  stars  of  the  fourth  and  fifth  magnitudes  which  opens 
toward  the  northeast.  The  Pawnee  Indians  called  it  the 
camp  circle,  and  it  is  not  difficult  to  imagine  that  the  stars 
represent  warriors  sitting  in  a  semicircle  around  a  central 
campfire. 

92.  Bootes  (The  Hunter).  —  Bootes  is  a  large  constel- 
lation lying  west  of  Corona  Borealis,  in  right  ascension  about 
14  hours,  and  extending  from  near  the  equator  to  within 
35°  of  the  pole.  It  always  can  be  easily  recognized  by  its 
bright  first-magnitude  star  Arcturus,  which  is  about  20° 
southwest  of  Corona  Borealis.  This 
star  is  a  deep  orange  in  color  and  is 
one  of  the  finest  stars  in  the  northern 
sky.  It  is  so  far  away  that  100 
years  are  required  for  its  fight  to 
come  to  the  earth,  and  in  radiating 
power  it  is  equivalent  to  more  than 
500  suns  fike  our  own. 

In  mythology  Bootes  is  represented 
as  leading  his  hunting  dogs  in  their 
pursuit  of  the  bear  across  the  sky. 

93.  Leo  (The  Lion).  —  Leo  Ues 
about  60°  west  of  Arcturus  and  is  the 
sixth  zodiacal  constellation.  It  is 
easily  recognized  by  the  fact  that  it  contains  7  stars  which 
form  the  outfine  of  a  sickle.  In  the  photograph,  Fig.  57,  only 
the  5  brightest  stars  are  shown.    The  most  southerly  star  of 


Fig.  57. — The  sickle  in  Leo, 
as  seen  when  it  is  on  the 
meridian. 


158        AN   INTRODUCTION   TO  ASTRONOMY     [ch.  v,  93 


Fig.  58.  —  The  Great  Andromeda  Nebula.     Photographed  by  Ritchey  vnth 
the  two-foot  reflector  of  the  Yerkes  Observatory. 


CH.  V,  95]  THE   CONSTELLATIONS  159' 

the  sickle  is  Regulus,  at  the  end  of  the  handle.  The  blade 
of  the  sickle  opens  out  toward  the  southwest.  One  of  the 
most  interesting  things  in  connection  with  this  constella- 
tion is  that  the  meteors  of  the  shower  which  occurs  about 
November  14  seem  to  radiate  from  a  point  within  the  blade 
of  the  sickle  (Art.  204). 

The  star  Regulus  is  at  the  heart  of  the  Nemean  Hon  which, 
according  to  classic  legends,  was  killed  by  Hercules  as  the 
first  of  his  twelve  great  labors. 

94.  Andromeda  (The  Woman  Chained).  —  Andromeda 
is  a  large  constellation  just  south  of  Cassiopeia.  It  contains 
no  first-magnitude  stars,  but  it  can  be  recognized  from  a 
line  of  3  second-magnitude  stars  extending  northeast  and 
southwest.  The  most  interesting  object  in  this  constellation 
is  the  Great  Andromeda  Nebula,  Fig.  58,  the  brightest 
nebula  in  the  sky.  It  is  about  15°  directly  south  of  Alpha 
Cassiopeise,  and  it  can  be  seen  without  difficulty  on  a  clear, 
moonless  night  as  a  hazy  patch  of  fight.  When  viewed 
through  a  telescope  it  fills  a  part  of  the  sky  nearly  2°  long  and 
1°  wide.  In  its  center  is  a  star  which  is  probably  variable. 
The  analysis  of  its  fight  with  the  spectroscope  seems  to  in- 
dicate that  it  is  composed  of  sofid  or  fiquid  material  sur- 
rounded by  cooler  gases.  It  has  been  suggested  that,  in- 
stead of  being  a  nebula,  it  may  be  an  aggregation  of  milfions 
of  Sims  comparable  to  the  Galaxy,  but  so  distant  from  us 
that  it  apparently  covers  an  insignificant  part  of  the  sky. 

95.  Perseus  (The  Champion).  —  Perseus  is  a  large  con- 
stellation in  the  Milky  Way  directly  east  of  Andromeda. 
Its  brightest  star,  Alpha,  is  in  the  midst  of  a  star  field  which 
presents  the  finest  spectacle  through  field  glasses  or  a  small 
telescope  in  the  whole  sky.  The  second  brightest  star  in 
this  constellation  is  the  earfiest  known  variable  star,  Algol 
(the  Demon).  Algol  is  about  9°  south  and  a  fittle  west  of 
Alpha  Persei,  and  varies  in  magnitude  from  2.2  to  3.4  in  a 
period  of  2.867  days.  That  is,  at  its  minimum  it  loses  more 
than  two  thirds  of  its  fight.     There  is  also  a  remarkable 


160        AN   INTRODUCTION   TO  ASTRONOMY     [ch.  v,  99 

double  cluster  in  this  constellation  about  10°  east  of  Alpha 
CassiopeisB. 

Algol,  together  with  the  little  stars  near  it,  is  the  Medusa's 
head  which  Perseus  is  supposed  to  carry  in  his  hand  and  which 
he  used  in  the  rescue  of  Andromeda.  He  is  said  to  have 
stirred  up  the  dust  in  heaven  in  his  haste,  and  it  now  ap- 
pears as  the  Milky  Way. 

96.  Atxriga  (The  Charioteer).  —  The  next  constellation 
east  of  Perseus  is  Auriga,  which  contains  the  great  first- 
magnitude  star  Capella.  Capella  is  about  40°  from  the 
Big  Dipper  and  nearly  in  a  fine  from  Delta  through  Alpha 
UrssB  Majoris.  It  is  also  distinguished  by  the  fact  that 
near  it  are  3  stars  ^known  as  The  Kids,  the  name  Capella 
meaning  The  She-goat.  It  is  receding  from  us  at  the  rate 
of  nearly  20  miles  per  second  and  its  distance  is  2,600,000 
times  that  of  the  earth  from  the  sun.  It  was  found  at  the 
Lick  Observatory,  in  1889,  to  be  a  spectroscopic  binary  with 
a  period  of  104.2  days.  The  computations  of  Maunder 
show  that  it  radiates  about  200  times  as  much  Ught  as  is 
given  out  by  the  sun. 

97.  Taurus  (The  Bull).  —  Taurus  is  southwest  of  Auriga 
and  contains  two  conspicuous  groups  of  stars,  the  Pleiades 
and  the  Hyades,  besides  the  brilliant  red  star  Aldebaran. 

Among  the  many  mythical  stories  regarding  this  constel- 
lation there  is  one  which  describes  the  bull  as  charging  down 
on  Orion.  According  to  a  Greek  legend,  Zeus  took  the  form 
of  a  bull  when  he  captured  Europa,  the  daughter  of  Agenor. 
While  playing  in  the  meadows  Avith  her  friends,  she  leaped 
upon  the  back  of  a  beautiful  white  bull,  which  was  Zeus 
himself  in  disguise.  He  dashed  into  the  sea  and  bore  her 
away  to  Crete.  Only  his  head  and  shoulders  are  visible  in 
the  sky  because,  when  he  swims,  the  rest  of  his  body  is 
covered  with  water. 

The  Pleiades  group.  Fig.  59,  consists  of  7  stars  in  the 
form  of  a  little  dipper  about  30°  southwest  of  Capella  and 
nearly  20°  south  of,  and  a  little  east  of,  Algol.     Six  of  them. 


CH.  V,  97] 


THE   CONSTELLATIONS 


161 


which  are  of  jhe  fourth  magnitude,  are  easily  visible  without 
optical  aid ;  'but  the  seventh,  which  is  near  the  one  at  the 
end  of  the  handle  in  the  dipper,  is  more  difficult.  There 
seems  to  have  been  considerable  difficulty  in  seeing  the  faint- 
est one  in  ancient  times,  for  it  was  frequently  spoken  of  as 


Fig.  59.  —  The  Pleiades.     Photographed  by  Wallace  at  the  Yerkes  Observatory. 


having  been  lost. '  There  is  no  difficulty  now,  however,  for 
people  with  good  eyes  to  see  it,  while  those  with  exceptionally 
keen  sight  can  see  10  or  11  stars. 

No  group  of  stars  in  all  the  sky  seems  to  have  attracted 
greater  popular  attention  than  the  Pleiades,  nor  tci  have 
been  mentioned  more  frequently,  not  only  in  the  classic 
writings  of  the  ancients,  but  also  in  the  stories  of  primitive 
peoples.  They  were  The  Seven  Sisters  of  the  Greeks,  The 
Many  Little  Ones  of  the  ancient  Babylonians,  The  Hen  and 
Chickens  of  the  peoples  of  many  parts  of  Eiu-ope,  The 
Little  Eyes  of  the  savage  tribes  of  the  South  Pacific  Islands, 
and  The  Seven  Brothers  of  some  of  the  tribes  of  North 
American  Indians.  They  cross  the  meridian  at  midnight 
in  November,  and  many  primitive  peoples  began  their  year 


162        AN   INTRODUCTION   TO  ASTRONOMY     [ch.  v,  97 

at  that  time.  It  is  said  that  on  the  exact  date,  November 
17,  no  petition  was  ever  presented  in  vain  to  the  kings  of 
ancient  Persia.  These  stars  had  an  important  relation  to 
the  reUgious  ceremonies  of  the  Aztecs,  and  certain  of  the 
Australian  tribes  held  dances  in  their  honor. 

Besides  the  7  stars  which  make  up  the  Pleiades  as  observed 
without  a  telescope,  there  are  at  least  100  others  in  the  group 
which  can  be  seen  with  a  small  instrument.  While  their 
distance  from  the  earth  is  not  known,  it  can  scarcely  be  less 
than  10,000,000  times  that  of  the  sun.  It  follows  that  these 
stars  are  apparently  small  only  because  they  are  so  remote. 
A  star  among  them  equal  to  the  sun  in  brilliancy  would  ap- 
pear to  us  as  a  telescopic  object  of  the  ninth  magnitude. 
The  larger  stars  of  the  group  are  at  least  from  100  to  200 
times  as  great  in  hght-giving  power  as  the  sim. 

About  8°  southeast  of  the  Pleiades  is  the  Hyades  group,  a 
cluster  of  small  stars  scarcely  less  celebrated  in  mythology. 
They  have  been  found  recently  to  constitute  a  cluster  of 
stars,  occupying  an  enormous  space,  all  of  which  move  in  the 
same  direction  with  almost  exactly  equal  speeds  (Art.  277). 
The  magnificent  scale  of  this  group  of  stars  is  quite  beyond 
imagination.  Individually  they  range  in  luminosity  from  5 
to  100  times  that  of  the  sun,  and  the  diameter  of  the  space 
which  they  occupy  is  more  than  2,000,000  times  the  dis- 
tance from  the  earth  to  the  sun. 

98.  Orion  (The  Warrior). — Southeast  of  Taurus  and 
directly  south  of  Auriga  is  the  constellation  Orion,  lying 
across  the  equator  between  the  fifth  and  sixth  hours  of  right 
ascension.  This  is  the  finest  region  of  the  whole  sky  for 
observation  without  a  telescope. 

The  legends  regarding  Orion  are  many  and  in  their  details 
conflicting.  But  in  all  of  them  he  was  a  giant  and  a  mighty 
hunter  who,  in  the  sky,  stands  facing  the  bull  (Taurus)  with 
a  club  in  his  right  hand  and  a  lion's  skin  in  his  left. 

About  7°  north  of  the  equator  and  15°  southeast  of  Al- 
debaran  is  the  ruddy  Betelgeuze.     About  20°  southwest  of 


CH.  V,  98]  THE   CONSTELLATIONS  163 

Betelgeuze  is  the  first-magnitude  star  Rigel,  a  magnificent 
object  which  is  at  least  2000  times  as  luminous  as  the  sun. 
About  midway  between  Betelgeuze  and  Rigel  and  almost 
on  the  equator  is  a  row  of  second-magnitude  stars  running 
northwest  and  southeast,  which  constitute  the  Belt  of  Orion, 
Fig.  60.  From  its  southern  end  another  row  of  fainter 
stars  reaches  off  to  the  southwest,  nearly  in  the  direction  of 


Pig.  60.  —  Orion.     Photographed  at  the  Yerkes  Observatory  (Hughes). 

Rigel.  These  stars  constitute  the  Sword  of  Orion.  The  cen- 
tral one  of  them  appears  a  little  fuzzy  without  a  telescope, 
and  with  a  telescope  is  found  to  be  a  magnificent  nebula. 
Fig.  61.  In  fact,  the  Great  Orion  Nebula  impresses  many 
observers  as  being  the  most  magnificent  object  in  the  whole 
heavens.  It  covers  more  than  a  square  degree  in  the  sky, 
and  the  spectroscope  shows  it  to  be  a  mass  of  glowing  gas 
whose  distance  is  probably  several  million  times  as  great  as 
that  to  the  sun,  and  whose  diameter  is  probably  as  great  as 


164       AN   INTRODUCTION   TO   ASTRONOMY      [ch.  v,  98 

the  distance  from  the  earth  to  the  nearest  star.  The  stars 
in  this  region  of  the  sky  are  generally  supposed  by  astronomers 
to  be  in  an  early  stage  of  their  development ;  most  of  them 


FiQ.  61.  —  The  Great  Orion  Nebula.     Photographed  by  Ritchey  with  the 
two-foot  reflector  of  the  Yerkes  Observatory. 

are  of  great  luminosity,  and  a  considerable  fraction  of  them 
are  variable  or  double. 


CH.  V,  100]  THE    CONSTELLATIONS  165 

99.  Canis  Major  (The  Greater  Dog).  —  The  constellation 
Canis  Major  is  southeast  of  Orion  and  is  marked  by  Sirius, 
the  brightest  star  in  the  whole  sky.  Sirius  is  almost  in  a 
line  with  the  Belt  of  Orion  and  a  little  more  than  20°  from  it. 
It  is  bluish  white  in  color  and  is  supposed  to  be  in  an  early 
stage  of  its  evolution,  though  it  has  advanced  somewhat  from 
the  condition  of  the  Orion  stars.  Sirius  is  comparatively 
near  to  us,  being  the  third  star  in  distance  from  the  sun. 
Nevertheless,  8.4  years  are  required  for  its  hght  to  come  to 
us,  and  its  distance  is  47,000,000,000,000  miles.  It  is  ap- 
proaching us  at  the  rate  of  5.6  miles  per  second ;  or,  rather,  it 
is  overtaking  the  svm,  for  the  solar  system  is  moving  in  nearly 
the  opposite  direction. 

f  he  history  of  Sirius  during  the  last  two  centuries  is  very 
interesting,  and  furnishes  a  good  illustration  of  the  value 
of  the  deductive  method  in  making  discoveries.  First, 
Halley  found,  in  1718,  that  Sirius  has  a  motion  with  respect 
to  fixed  reference  points  and  lines ;  then,  a  httle  more  than 
a  century  later,  Bessel  found  that  this  motion  is  slightly 
variable.  He  inferred  from  this,  on  the  basis  of  the  laws  of 
motion,  that  Sirius  and  an  unseen  companion  were  traveling 
around  their  common  center  of  gravity  which  was  moviiig 
with  uniform  speed  in  a  straight  line.  This  companion 
actually  was  discovered  by  Alvan  G.  Clark,  in  1862,  while 
adjusting  the  18-inch  telescope  now  of  the  Dearborn  Ob- 
servatory, at  Evanston,  111.  The  distance  of  the  two  stars 
from  each  other  is  1,800,000,000  miles,  and  they  complete 
a  revolution  in  48.8  years.  The  combined  mass  of  the 
two  stars  is  about  3.4  times  that  of  the  sun.  The  larger 
star  is  only  about  twice  as  massive  as  its  companion  but  is 
20,000  times  brighter;  together  they  radiate  48  times  as 
much  light  as  is  emitted  by  the  sun. 

100.  Canis  Minor  (The  Lesser  Dog).  —  Canis  Minor  is 
directly  east  of  Orion  and  is  of  particular  interest  in  the 
present  connection  because  of  its  first-magnitude  star  Pro- 
cyon,  which  is  about  25°  east  and  just  a  Uttle  south  of  Betel- 


166       AN   INTRODUCTION   TO  ASTRONOMY     [ch,  v,  100 

geuze.  The  history  of  this  star  is  much  the  same  as  that  of 
Sirius,  the  fainter  companion  having  been  discovered  in 
1896  by  Schaeberle  at  the  Lick  Observatory.  The  period  of 
revolution  of  Procyon  and  its  companion  is  39  years,  its 
distance  is  a  little  greater  than  that  of  Sirius,  its  combined 
mass  is  about  1.3  that  of  the  sun,  and  its  luminosity  is  about 
10  times  that  of  the  sun.  If  the  orbits  of  such  systems  as 
Sirius  and  Procyon  and  their  fainter  companions  were  edge- 
wise to  the  earth,  the  brighter  components  would  be  regu- 
larly ecUpsed  and  they  would  be  variable  stars  of  the  Algol 
type  (Art.  288),  though  with  such  long  periods  and  short 
times  of  eclipse  that  their  variabihty  would  probably  not  be 
discovered. 

.  101.  Gemini  (The  Twins).  —  Gemini  is  the  fourth  zodiacal 
constellation  and  lies  directly  north  of  Canis  Minor.  It  has 
been  known  as  "  The  Twins  "  from  the  most  ancient  times 
because  its  two  principal  stars.  Castor  and  Pollux,  are 
almost  ahke  and  only  4°. 5  apart.  These  stars  are  about 
25°  north  of  Procyon,  and  Castor  is  the  more  northerly  of 
the  two.  Castor  is  a  double  star  which  can  be  separate4  by 
a  small  telescope.  In  1900  B61opolsky,  of  Pulkowa,  found 
that  its  fainter  companion  is  a  spectroscopic  binary  with  a 
period  of  2.9  days.  In  1906  Curtis,  of  the  Lick  Observatory, 
found  that  the  brighter  companion  is  also  a  spectroscopic 
binary  with  a  period  of  9.2  days.  Thus  this  star,  instead 
of  being  a  single  object  as  it  appears  to  be  without  telescopic 
and  spectroscopic  aid,  is  a  system  of  four  suns.  The  two 
pairs  revolve  about  the  common  center  of  gravity  of  the  four 
stars  in  a  long  period  which  probably  lies  between  250  and 
2000  years. 

Castor  is  called  Alpha  Geminorum,  because  probably  in 
ancient  times  it  was  a  little  brighter  than,  or  at  least  as  bright 
as,  Pollux.     Now  Pollux  is  a  Httle  brighter  than  Castor. 

About  10°  southeast  of  Pollux  is  the  large  open  Prssepe 
(The  Beehive)  star  cluster  which  can  be  seen  on  a  clear, 
moonless  night  without  a  telescope. 


CH.  V,  102]  THE   CONSTELLATIONS  167 

102.  On  Becoming  Familiar  with  the  Stars.  —  The  dis- 
cussion of  the  constellations  will  be  closed  here,  not  because 
all  have  been  described,  or,  indeed,  any  one  of  them  ade- 
quately, but  because  enough  has  been  said  to  show  that  the 
sky  is  full  of  objects  of  interest  which  can  be  found  and  en- 
joyed with  very  httle  optical  aid.  The  reader  is  expected 
to  observe  all  the  objects  which  have  been  described,  so  far 
as  the  time  of  year  and  the  instrumental  help  at  his  coih- 
mand  will  permit.  If  he  does  this,  the  whole  subject  will 
have  a  deeper  and  more  lively  interest,  and  it  will  be  a  pleas- 
ure to  make  constant  appeals  to  the  sky  to  verify  statements 
and  descriptions. 

The  general  features  of  the  constellations  are  very  simple, 
but  the  whole  subject  cannot  be  mastered  in  an  evening. 
One  should  go  over  it  several  times  with  no  greater  optical 
aid  than  that  furnished  by  a  field  glass. 

VIII.  QUESTIONS 

1.  Show  why  about  22,000  plates  will  be  required  to  photograph 
'the  whole  sky  as  described  in  Art.  76. 

2.  Find  the  brightness  of  the  stars  in  Table  I  compared  to  that 
of  a  flrst-magnitude  star. 

3.  Find  the  amount  of  light  received  from  the  sun  compared  to 
that  received  from  a  first-magnitude  star. 

4.  Take  the  amount  of  light  received  from  a  first-magnitude 
star  as  unity,  and  compute. the  amount  of  light  received  from  each 
of  the  first  six  magnitudes  (Table  II). 

5.  If  the  ratio  of  the  number  of  stars  from  one  magnitude  to  the 
next  continued  the  same  as  it  is  in  Table  II,  how  many  stars  would 
there  be  in  the  first  20  magnitudes  ? 

6.  At  what  time  of  the  year  is  the  most  northerly  part  of  the 
Milky  Way  on  the  meridian  at  8  p.m.  ?  What  are  its  altitude  and 
azimuth  at  that  time  ? 

7.  What  constellations  are  within  two  hours  of  the  meridian  at 
8  P.M.  to-night  ?    Identify  them. 

8.  If  Lyra  is  visible  at  a  convenient  hour,  test  your  eyes  on 
Epsilon  LyrsB. 

9.  If  Leo  is  visible  at  a  convenient  hour,  test  your  eyes  by  find- 
ing which  star  in  the  sickle  has  a  very  faint  star  near  it. 


168      AN  INTKODUCTION   TO   ASTRONOMY     [ch.  v,  102 

10.  If  Andromeda  is  visible  at  a  convenient  hour,  find  the  great 
nebula. 

11.  How  many  stars  can  you  see  in  the  bowl  of  the  Big  Dipper  ? 

12.  If  Perseus  is  visible  at  a  convenient  hour,  identify  Algol  and 
verify  its  variabihty. 

13.  How  many  of  the  Pleiades  can  you  see  ? 

14.  If  Orion  is  visible  at  a  convenient  hour,  identify  the  Belt  and 
Sword  and  notice  that  the  great  nebula  looks  hke  a  fuzzy  star. 


CHAPTER  VI 
TIME 

103.  Definitions  of  equal  Intervals  of  Time.  —  It  is  impos- 
sible to  give  a  definition  of  time"  in  terms  which  are  simpler 
and  better  understood  than  the  word  itself ;  but  it  is  profit- 
able to  consider  what  it  is  that  determines  the  length  of  an 
interval  of  time.  The  subject  may  be  considered  from  the 
standpoint  of  the  intellectual  experience  of  the  individual, 
which  varies  greatly  from  time  to  time  and  which  may  differ 
much  from  that  of  another  person,  or  it  may  be  treated  with 
reference  to  independent  physical  phenomena. 

Consider  first  the  definition  of  the  length  of  an  interval 
of  time  or,  rather,  the  equality  of  two  intervals  of  time, 
from  the  psychological  point  of  view.  If  a  person  has  had  a 
number  of  intellectual  experiences,  he  is  not  only  conscious 
that  they  were  distinct,  but  he  has  them  arranged  in  his  mem- 
ory in  a  perfectly  definite  order.  When  he  recalls  them  and 
notes  their  distinctness,  number,  and  order,  he  feels  that  they 
have  occurred  in  time ;  that  is,  he  has  the  perception  of  time. 
An  interval  in  which  a  person  has  had  many  and  acute 
intellectual  experiences  seems  long;  and  two  intervals  of 
time  are  of  equal  length,  psychologically,  when  the  individual 
has  had  in  them  an  equal  number  of  equally  intense  intellec- 
tual experiences.  For  example,  in  youth  when  most  of  life's 
experiences  are  new  and  wonderful,  the  months  and  the 
years  seem  to  pass  slowly ;  on  the  other  hand,  with  increas- 
ing age  when  life  reduces  largely  to  routine,  the  years  slip 
away  quickly.  Or,  to  take  an  illustration  within  the  range 
of  the  experience  of  many  who  are  still  young,  a  month  of 
travel,  or  the  first  month  in  college,  seems  longer  than  a  whole 
year  in  the  accustomed  routine  of  preparatory  school  life. 

169 


170     AN  INTRODUCTION   TO  ASTRONOMY     [ch.  vi,  103 

It  follows  from  these  considerations  that  the  true  measure  of 
the  length  of  the  life  of  an  individual  from  the  psychological 
point  of  view,  which  is  the  one  in  which  he  has  greatest  inter- 
est as  a  thinking  being,  is  the  number,  variety,  and  intensity 
of  his  intellectual  experiences.  A  man  whose  life  has  been 
full,  who  has  become  acquainted  with  the  world's  history, 
who  is  familiar  with  the  wonders  of  the  universe,  who  has 
read  and  experienced  again  the  finest  thoughts  of  the  best 
minds  of  all  ages,  who  has  seen  many  places  and  come  into, 
contact  with  many  men,  and  who  has  originated  ideas  and 
initiated  intellectual  movements  of  his  own,  has  lived  a 
long  life,  however  few  may  have  been  the  number  of  revolu- 
tions of  the  earth  around  the  sun  since  he  was  born. 

But  since  men  must  deal  with  one  another,  it  is  important 
to  have  some  definition  of  the  equality  of  intervals  of  time 
that  will  be  independent  of  their  varying  intellectual  life. 
The  definition,  or  at  least  its  consequences,  must  be  capable 
of  being  applied  at  any  time  or  place,  and  it  must  not  dis- 
agree too  radically  with  the  psychological  definition.  Such 
a  definition  is  given  by  the  first  law  of  motion  (Art.  40),  or 
rather  a  part  of  it,  which  for  present  purposes  will  be  reworded 
as  follows : 

Two  intervals  of  time  are  equal,  by  definition,  if  a  moving 
body  which  is  subject  to  no  forces  passes  over  equal  distances  in 
them.  It  is  estabUshed  by  experience  that  it  makes  no 
difference  what  moving  body  is  used  or  at  what  rate  it  moves, 
for  they  all  give  the  same  result. 

104.  The  Practical  Measure  of  Time.  —  A  diflSculty 
with  the  first  law  of  motion  and  the  resulting  definition  of 
equal  intervals  of  time  arises  from  the  fact  that  it  is  impos- 
sible to  find  a  body  which  is  absolutely  uninfluenced  by 
exterior  forces.  Therefore,  instead  of  using  the  law  itself, 
one  of  its  indirect  consequences  is  employed.  It  follows 
from  this  law,  together  with  the  other  laws  of  motion,  that 
a  solid,  rotating  sphere  which  is  subject  to  no  exterior  forces 
turns  at  a  uniform  rate.    There  is  no  rotating  body  which 


CH.  VI,  105]  TIIkfE  171 

is  not  subject  to  at  least  the  attraction  of  other  bodies ;  but 
the  simple  attraction  of  an  exterior  body  has  no  influence 
on  the  rate  of  rotation  of  a  sphere  which  is  perfectly  sohd. 
Therefore  the  earth  rotates  at  a  uniform  rate,  according  to 
the  definition  of  uniformity  implied  in  the  first  law  of  motion, 
except  for  the  sUght  and  altogether  negligible  modifying  in- 
fluences which  were  enmnerated  in  Art.  45,  and  hence  can 
be  used  for  the  measurement  of  time. 

If  the  rotation  of  the  earth  is  to  be  used  in  the  measure- 
ment of  time,  it  is  only  necessary  to  determine  in  some  way 
the  angle  through  which  it  turns  in  any  interval  under  con- 
sideration. This  can  be  done  by  observations  of  the  position 
of  the  meridian  with  reference  to  the  stars.  Since  the  stars 
are  extremely  far  away  and  do  not  move  appreciably  with 
respect  to  one  another  in  so  short  an  interval  as  a  day,  the 
rotation  of  the' earth  can  be  measured  by  reference  to  any 
of  them.  Let  it  be  remembered  that,  though  the  rate  of 
the  rotation  of  the  earth  is  subject  to  some  possible  slight 
modifications,  its  uniformity  is  far  beyond  that  of  any  clock 
ever  made. 

105.  Sidereal  Time.  —  Sidereal  time  is  the  time  defined 
by  the  rotation  of  the  earth  with  respect  to  the  stars.  A 
sidereal  day  is  the  interval  between  the  passage  of  the 
meridian,  in  its  eastward  motion,  across  a  star  and  its  next 
succeeding  passage  across  the  same  star.  Since  the  earth 
rotates  at  a  uniform  rate,  all  sidereal  days  are  of  the  same 
length.  The  sidereal  day  is  divided  into  24  sidereal  hours, 
which  are  numbered  from  1  to  24,  the  hours  are  divided  into 
60  minutes,  and  the  minutes  into  60  seconds.  The  sidereal 
time  of  a  given  place  on  the  earth  is  zero  when  its  meridian 
crosses  the  vernal  equinox. 

Since  the  definition  of  sidereal  time  depends  upon  the 
meridian  of  the  observer,  it  follows  that  all  places  on  the 
earth  having  the  same  longitude  have  the  same  sidereal 
time,  and  that  those  having  different  longitudes  have  dif- 
ferent sidereal  time.    It  follows  from  the  uniformity  of  the 


172     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  vi,  105 


earth's  rotation  that  equal  intervals  of  sidereal  time  are 
equal  according  to  the  first  law  of  motion. 

106.  Solar  Time.  —  Solar  time  is  defined  by  the  rotation 
of  the  earth  with  respect  to  the  sun.  A  solar  day  is  the 
interval  of  time  between  the  passage  of  a  meridian  across 
the  center  of  the  sun  and  its  next  succeeding  passage  across 
the  center  of  the  sun.  Since  the  sun  apparently  moves 
eastward  among  the  stars,  a  solar  day  is  longer  than  the 
sidereal  day.  The  sun  makes  an  apparent  revolution  of  the 
heavens  in  365  days,  and  therefore,  since  the  circuit  of  the 
heavens  is  360°,  it  moves  eastward  on  the  average  a  little 
less  than  1°  a  day.  The  earth  turns  15°  in  1  hour,  and  1° 
in  4  minutes,  from  which  it  follows  that  the  solar  day  is 
nearly  4  minutes  longer  on  the  average  than  the  sidereal  day. 

107.  Variations  in  the  Lengths  of  Solar  Days.  —  If  the 
apparent  motion  of  the  sun  eastward  among  the  stars  were 

uniform,  each 
solar  day  would 
be  longer  than 
the  sidereal  day 
by  the  same 
amount;  and 
since  the  sidereal 
days  are  all  of 
equal  length,  the 
solar  days  also 
would  all  be  of 
equal  length. 
But  the  east- 
ward apparent 
motion    of     the 

sun  is  somewhat  variable  because  of  two  principal  reasons, 

which  will  now  be  explained. 

The  earth  moves  in  its  elliptical  orbit  around  the  sun  in 

such  a  way  that  the  law  of  areas  is  fulfilled.     The  angular 

distance  the  sun  appears  to  move  eastward  among  the 


Fig.  62.  —  Solar  days  are  longer  than  sidereal  days. 


CH.  VI,  107] 


TIME 


173 


stars  equals  the  angular  distance  the  earth  moves  forward 
in  its  orbit.  This  is  made  evident  from  Fig.  62,  in  which 
El  represents  the  position  of  the  earth  when  it  is  noon  at  A. 
At  the  next  noon  at  A,  solar  time,  the  earth  has  moved  for- 
ward in  its  orbit  through  the  angle  EiSE^  (of  course  the  dis- 
tance is  greatly  exaggerated).  Suppose  that  when  the  earth 
is  at  El  the  direction  of  a  star  is  EiS.  When  the  earth  is  at 
Ei,  the  same  direction  is  E^S'.  The  sun  has  apparently 
moved  through  the  angle  S'E^S,  which  equals  E^SEi. 

Since  the  earth  moves  in  its  orbit  in  accordance  with  the 
law  of  areas,  its  angular  motion  is  fastest  when  it  is  nearest 


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JAN.       FEB.       AMR     APR      MAY     JUNE   JULY     AUG     SEPT.       OCT      NOli     DEC     JAN. 

Fig.  63.  —  Length  of  solar  days.    Broken  line  gives  effects  of  eccentricity ; 
dotted  line,  the  inclination ;  full  Une,  the  combined  effects. 


the  sun.  Consequently,  when  the  earth  is  at  its  perihelion 
the  sun's  apparent  motion  eastward  is  fastest,  and  the  solar 
days,  so  far  as  this  factor  alone  is  concerned,  are  then  the 
longest.  The  earth  is  at  its  perihelion  point  about  the  first 
of  January  and  at  its  aphelion  point  about  the  first  of  July. 
Consequently,  the  time  from  noon  to  noon,  so  far  as  it 
depends  upon  the  eccentricity  of  the  earth's  orbit,  is  longest 
about  the  first  of  January  and  shortest  about  the  first  of 
July.  The  lengths  of  the  solar  days,  so  far  as  they  depend 
upon  the  eccentricity  of  the  earth's  orbit,  are  shown  by  the 
broken  line  in  Fig.  63. 

The  second  important  reason  why  the  solar  days  vary 
in  length  is  that  the  sun  moves  eastward  along  the  echptic 
and  not  along  the  equator.  For  simpUcity,  neglect  the 
eccentricity  of  the  earth's  orbit  and  the  lack  of  uniformity  of 
the  angular  motion  of  the  sun  along  the  ecliptic.     Consider 


174     AN   INTRODUCTION   TO  ASTRONOMY     [ch.  vi,  107 

the  time  when  the  sun  is  near  the  vernal  equinox.  Since 
the  ecUptic  intersects  the  equator  at  an  angle  of  23°.5,  only- 
one  component  of  the  sun's  motion  is  directly  eastward. 
However,  the  reduction  is  somewhat  less  than  might  be 
imagined  for  so  large  an  inclination  and  amounts  to  only 
about  10  per  cent.  When  the  sun  is  near  the  autimmal 
equinox  the  situation  is  the  same  except  that,  at  this  time, 
one  component  of  the  sun's  motion  is  toward  the  south. 
At  these  two  times  in  the  year  the  sun's  apparent  motion 
eastward  is  less  than  it  would  otherwise  be,  and,  conse- 
quently, the  solar  days  are  shorter  than  the  average.  At  the 
solstices,  midway  between  these  two  periods,  the  sun  is 
moving  ^approximately  along  the  arcs  of  small  circles  23°.5 
from  the  equator,  and  its  angular  motion  eastward  is  cor- 
respondingly faster  than  the  average.  Therefore,  so  far 
as  the  inchnation  of  the  echptic  is  concerned,  the  solar  days 
are  longest  about  December  21  and  June  21,  and  shortest 
about  March  21  and  September  23.  The  lengths  of  the 
solar  flays,  so  far  as  they  depend  upon  the  inchnation  of 
the  echptic,  are  shown  by  the  dotted  curve  in  Fig.  63. 

Now  consider  the  combined  effects  of  the  eccentricity  of 
the  earth's  orbit  and  the  incliijation  of  the  ecUptic  on  the 
lengths  of  the  solar  days.  Of  these  two  influences,  the 
inchnation  of  the  echptic  is  considerably  the  more  impor- 
tant. On  the  first  of  January  they  both  make  the  solar  day 
longer  than  the  average.  At  the  vernal  equinox  the  eccen- 
tricity has  only  a  shght  effect  on  the  length  of  the  solar  day, 
while  the  obhquity  of  the  ecliptic  makes  it  shorter  than  the 
average.  On  June  21  the  effect  of  the  eccentricity  is  to 
make  the  solar  day  shorter  than  the  average,  while  the  effect 
of  the  obliquity  of  the  echptic  is  to  make  it  longer  than  the 
average.  At  the  autumnal  equinox  the  eccentricity  has 
only  a  shght  importance  and  the  obhquity  of  the  echptic 
makes  the  solar  day  shorter  than  the  average. 

The  two  influences  together  give  the  following  result: 
The  longest  day  in  the  year,  from  noon  to  noon  by  the  sun. 


CH.  VI,  108]  TIME  175 

is  about  December  22,  after  which  the  solar  day  decreases 
continually  in  length  until  about  the  26th  of  March;  it 
then  increases  in  length  until  about  June  21 ;  then  it  decreases 
in  length  until  the  shortest  day  in  the  year  is  reached  on 
September  17;  and  then  it  increases  in  length  continually 
until  December  22.  On  December  22  the  solar  day  is  about 
4  minutes  and  26  seconds  of  mean  solar  time  [Art.  108] 
longer  than  the  sidereal ;  on  March  26  it  is  3  minutes  and 
38  seconds  longer ;  on  June  21  it  is  4  minutes  9  seconds 
longer ;  and  on  September  17  it  is  3  minutes  and  35  seconds 
longer.  The  combined  results  are  shown  by  the  full  Hne  in 
Fig.  63.  The  difference  in  length  between  the  longest  and 
the  shortest  day  in  the  year  is,  therefore,  about  51  seconds  of 
mean  solar  time.  While  this  difference  for  most  purposes 
is  not  important  in  a  single  day,  it  accumulates  and  gives 
rise  to  what  is  known  as  the  equation  of  time  (Art.  109). 

It  might  seem  that  it  would  be  sensible  for  astronomers  to 
neglect  the  differences  in  the  lengths  of  the  solar  days, 
especially  as  the  change  in  length  from  one  day  to  the  next 
is  very  small.  Only  an  accurate  clock  would  show  the  dis- 
parity in  their  lengths,  and  their  shght  differences  would  be 
of  no  importance  in  ordinary  affairs.  But  if  astronomers 
should  use  the  rotation  of  the  earth  with  respect  to  the 
sun  as  defining  equal  intervals  of  time,  they  would  be 
emplojdng  a  varying  standard  and  they  would  find  apparent 
irregularities  in  the  revolution  of  the  earth  and  in  all  other 
celestial  motions  which  they  could  not  bring  under  any  fixed 
laws.  This  illustrates  the  extreme  sensitiveness  of  astro- 
nomical theories  to  even  slight  errors. 

108.  Mean  Solar  Time.  —  Since  the  ordinary  activities 
of  mankind  are  dependent  largely  upon  the  period  of  day- 
light, it  is  desirable  for  practical  purposes  to  have  a  unit  of 
time  based  in  some  way  upon  the  rotation  of  the  earth  with 
respect  to  the  sun.  On  the  other  hand,  it  is  undesirable  to 
have  a  unit  of  variable  length.  Consequently,  the  mean 
solar  day,  which  has  the  average  length  of  all  the  solar  days 


176     AN   INTRODUCTION   TO  ASTRONOMY     [oh.  vi,  108 

of  the  year,  is  introduced.  In  sidereal  time  its  length  is 
24  hours,  3  minutes,  and  56.555  seconds. 

The  mean  solar  day  is  divided  into  24  mean  solar  hours, 
the  hours  into  60  mean  solar  minutes,  and  the  minutes  into 
60  mean  solar  seconds.  These  are  the  hours,  minutes,  and 
seconds  in  common  use,  and  ordinary  timepieces  are  made 
to  keep  mean  solar  time  as  accurately  as  possible.  It  would 
be  very  difficult,  if  not  impossible,  to  construct  a  clock  that 
would  keep  true  solar  time  with  any  high  degree  of  precision. 

109.  The  Equation  of  Time.  —  The  difference  between  the 
true  solar  time  arid  the  mean  solar  time  of  a  place  is  called 
the  equation  of  time.  It  is  taken  with  such  an  algebraic  sign 
that,  when  it  is  added  to  the  mean  solar  time,  fhe  true  solar 
time  is  obtained. ' 

The  date  on  which  noon  by  mean  solar  time  and  true  solar 
time  shall  coincide  is  arbitrary,  but  it  is  so  chosen  that  the 


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JAN.    ne.    Mw.  APR   mr.  jum  mly  auo.  sept,   ixt    -mx  des-   jah 

Fig.  64.  —  The  equation  of  time. 


differences  between  the  times  in  the  two  systems  shall  be 
as  small  as  possible.  On  the  24th  of  December  the  equation 
of  time  is  zero.  It  then  becomes  negative  and  increases 
numerically  until  February  11,  when  it  amounts  to  about 
— 14  minutes  and  25  seconds ;  it  then  increases  and  passes 
through  zero  about  April  15,  after  which  it  becomes  positive 
and  reaches  a  value  of  3  minutes  48  seconds  on  May  14; 
it  then  decreases  and  passes  through  zero  on  June  14  and 
becomes  —  6  minutes  and  20  seconds  on  July  26 ;    it  then 

'  This  is  the  present  practice  of  the  American  Ephemeris  and  Nautical 
Almanac  ;  it  was  formerly  the  opposite. 


CH.  VI,  110]  ,    TIME  177 

increases  and  passes  through  zero  on  September  1  and 
becomes  16  minutes  and  21  seconds  on  November  2,  after 
which  it  continually  decreases  until  December  24.  The 
results  are  given  graphically  iii  Fig.  64.  The  dates  may 
vary  a  day  or  two  from  those  given  because  of  the  leap  year, 
and  the  amounts  by  a  few  seconds  because  of  the  shifting  of 
the  dates. 

Some  interesting  results  follow  from  the  equation  of  time. 
For  example,  on  December  24  the  equation  of  time  is  zero, 
but  the  solar  day  is  about  30  seconds  longer  than  the  mean 
solar  day.  Consequently,  the  next  day  the  sun  will  be  about 
30  seconds  slow ;  that  is,  noon  by  the  mean  solar  clock  has 
shifted  about  30  seconds  with  respect  to  the  sun.  As  the 
sun  has  just  passed  the  winter  solstice,  the  period  from  sun- 
rise to  sunset  for  the  northern  hemisphere  of  the  earth  is 
slowly  increasing,  the  exact  amount  depending  upon  the 
latitude.  For  latitude  40°  N.  the  gain  in  the  forenoon  result- 
ing from  the  earlier  rising  of  the  sun  is  less  than  the  loss 
from  the  shifting  of  the  time  of  the  noon.  Consequently, 
almanacs  will  show  that  the  forenoons  are  getting  shorter 
at  this  time  of  the  year,  although  the  whole  period  between 
sunrise  and  sunset  is  increasing.  The  difference  in  the 
lengths  of  the  forenoons  and  afternoons  may  accumulate 
until  it  amounts  to  nearly  half  an  hour. 

110.  Standard  Time.  —  The  mean  solar  time  of  a  place 
is  called  its  local  time.  All  places  having  the  same  longitude 
have  the  same  local  time,  but  places  having  different  longi- 
tudes have  different  local  times.  The  circumference  of  the 
earth  is  nearly  25,000  miles  and  15°  correspond  to  a  difference 
of  one  hour  in  local  time.  Consequently,  at  the  earth's 
equator,  17  miles  in  longitude  give  a  difference  of  about  one 
minute  in  local  time.  In  latitudes  40°  to  45°  north  or  south 
13  to  12  miles  in  longitude  give  a  difference  of  one  minute 
in  local  time. 

If  every  place  along  a  railroad  extending  east  and  west 
should  keep  its  own  local  time,  there  would  be  endless  con- 


178     AN   INTRODUCTION   TO  ASTRONOMY     [ch.  vi,  110 

fusion  and  great  danger  in  running  trains.  In  order  to  avoid 
these  difficulties,  it  has  been  agreed  that  all  places  whose 
local  times  do  not  differ  more  than  half  an  hour  from'  that  of 
some  convenient  meridian  shall  use  the  local  time  of  that 
meridian.  Thus,  while  the  extreme  difference  in  local  time 
of  places  using  the  local  time  of  the  same  meridian  may  be 
about  an  hour,  neither  of  them  differs  more  than  about  half 
an  hour  from  its  standard  time.  In  this  manner  a  strip  of 
country  about  750  miles  wide  in  latitudes  35°  to  45°  uses 
the  same  time,  and  the  next  strip  of  the  same  width  an  hour 
different,  and  so  on.  The  local  time  of  the  standard  meridian 
of  each  strip  is  the  standard  time  of  that  strip. 

At  present  standard  time  is  in  use  in  nearly  every  civiUzed 
part  of  the  earth.  The  United  States  and  British  America 
are  of  such  great  extent  in  longitude  that  it  is  necessary  to 
use  four  hours  of  standard  time.  The  eastern  portion  uses 
what  is  called  Eastern  Time.  It  is  the  local  time  of  the 
meridian  5  hours  west  of  Greenwich.  This  meridian  runs 
through  Philadelphia,'  and  in  this  city  local  time  and  standard 
time  are  identical.  At  places  east  of  this  meridian  it  is  later 
by  local  time  than  by  standard  time,  the  difference  being 
one  minute  for  12  or  13  miles.  At  places  west  of  this  meridian, 
but  in  the  Eastern  Time  division,  it  is  earlier  by  local  time 
than  by  standard  time.  The  next  division  to  the  westward 
is  called  Central  Time.  It  is  the  local  time  of  the  meridian 
6  hours  west  of  Greenwich,  which  passes  through  St.  Louis. 
The  next  time  division  is  called  Mountain  Time.  It  is  the 
local  time  of  the  meridian  7  hours  west  of  Greenwich.  This 
meridian  passes  through  Denver.  The  last  time  division 
is  called  Pacific  Time.  It  is  the  local  time  of  the  meridian 
8  hours  west  of  Greenwich.  This  meridian  passes  about  100 
miles  east  of  San  Francisco. 

If  the  exact  divisions  were  used,  the  boundaries  between 
one  time  division  and  the  next  would  be  7°.5  east  and  west  of 
the  standard  meridian.  As  a  matter  of  fact,  the  boundaries 
are   quite  irregular,    depending  upon   the   convenience  of 


CH.  VI,  111] 


TIME 


179 


railroads,  and  they  are  frequently  somewhat  altered.  The 
change  in  time  is  nearly  always  made  at  the  end  of  a  railway 
division;  for,  obviously,  it  would  be  unwise  to  have  rail- 
road time  change  during  the  run  of  a  given  train  'crew.  As 
a  result  the  actual  boundaries  of  the  several  time  divisions 
are  quite  irregular  and  vary  in  many  cases^  radically  from  the 


Fig.  65.  —  Standard  time  divisions  in  the  United  States. 


ideal  standard  divisions.     Moreover,  many  towns  near  the 
borders  of  the  time  zones  do  not  use  standard  time. 

111.  The  Distribution  of  Time.  —  The  accurate  deter- 
mination of  time  and  its  distribution  are  of  much  impor- 
tance. There  are  several  methods  by  which  time  may  be 
determined,  but  the  one  in  common  use  is  to  observe  the 
transits  of  stars  across  the  meridian  and  thus  to  obtain  the 
sidereal  time.  From  the  mathematical  theory  of  the  earth's 
motion  it  is  then  possible  to  compute  the  mean  solar  time. 
It  might  be  supposed  that  it  would  be  easier  to  find  mean 
solar  time  by  observing  the  transit  of  the  sun  across  the 
meridian,  but  this  is  not  true.     In  the  first  place,  it  is  much 


180     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  vi,  111 

more  difficult  to  determine  the  exact  time  of  the  transit 
of  the  sun's  center  than  it  is  to  determine  the  time  of  the 
transit  of  a  star ;  and,  in  the  second  place,  the  sun  crosses  the 
meridian  but  once  in  24  hours,  while  many  stars  may  be 
observed.  In  the  third  place,  observations  of  the  sun  give 
true  solar  time  instead  of  mean  solar  time,  and  the  com- 
putation necessary  to  reduce  from  one  to  the  other  is  as  diffi- 
cult as  it  is  to  change  from  sidereal  time  to  mean  solar  time. 
It  remains  to  explain  how  time  is  distributed  from  the 
places  where  the  observations  are  made.  In  most  countries 
the  time  service  is  under  the  control  of  the  government, 
and  the  time  signals  are  sent  out  from  the  national  observa- 
tory. For  example,  in  the  United  States,  the  chief  source 
of  time  for  railroads  and  commercial  purposes  is  the  Naval 
Observatory,  at  Georgetown  Heights,  Washington,  D.C. 
There  are  three  high-grade  clocks  keeping  standard  time 
at  this  observatory.  Their  errors  are  found  from  observations 
of  the  stars ;  and  after  applying  corrections  for  these  errors, 
the  mean  of  the  three  clocks  is  taken  as  giving  the  true 
standard  time  for  the  successive  24  hours.  At  5  minutes 
before  noon.  Eastern  Time,  the  Western  Union  Telegraph 
Company  and  the  Postal  Telegraph  Company  suspend  their 
ordinary  business  and  throw  their  Unes  into  electrical  con- 
nection with  the  standard  clock  at  the  Naval  Observatory. 
The  connection  is  arranged  so  that  the  sounding  key  makes 
a  stroke  every  second  during  the  5  minutes  preceding  noon 
except  the  twenty-ninth  second  of  each  minute,  the  last  5 
seconds  of  the  fourth  minute,  and  the  last  10  seconds  of  the 
fifth  minute.  This  gives  many  opportunities  of  determin- 
ing the  error  of  a  clock.  To  simphfy  matters,  clocks  are 
connected  so  as  to  be  automatically  regulated  by  these 
signals,  and  there  are  at  present  more  than  30,000  of  them 
in  use  in  this  country.  The  time  signals  are  sent  out  from 
the  Naval  Observatory  with  an  error  usually  less  than  0.2 
of  a  second;  but  frequently  this  is  considerably  increased 
when  a  systefai  of  relays  must  be  used  to  reach  great  distances. 


CH.  VI,  113]  TIME  181 

These  noon  signals  also  operate  time  balls  in  18  ports  in 
the  United  States.  This  device  for  furnishing  time,  chiefly 
to  boat  captains,  consists  of  a  large  ball  which  is  dropped  at 
noon.  Eastern  Time,  froOi  a  considerable  height  at  con- 
spicuous points,  by  means  of  electrical  connection  with  the 
Naval  Observatory. 

Time  for  the  extreme  western  part  of  the  United  States 
is  distributed  from  the  Mare  Island  Navy  Yard  in  Cahf ornia ; 
and  besides,  a  number  of  college  observatories  have  been 
furnishing  time  to  particular  railroad  systems.  Naturally 
most  observatories  regularly  determine  time  for  their  own 
use,  though  with  the  accurate  distribution  of  time  from 
Washington  the  need  for  this  work  is  disappearing  except 
in  certain  special  problems  of  star  positions. 

112.  Civil  and  Astronomical  Days.  —  The  civil  day  begins 
at  midnight,  for  then  business  is  ordinarily  suspended  and 
the  date  can  be  changed  with  least  inconvenience.  The 
astronomical  day  of  the  same  date  begins  at  noon,  12  hours 
later ;  because,  if  the  change  were  made  at  midnight,  astron- 
omers might  find  it  necessary  to  change  the  date  in  the 
midst  of  a  set  of  observations.  It  is  true  that  many  observa- 
tions of  the  sun  and  some  other  bodies  are  made  in  the  day- 
time, but  of  course  most  observational  work  is  done  at  night. 
The  hours  of  the  astronomical  day  are  numbered  up  to  24, 
just  as  in  the  case  of  sidereal  time.  ' 

113.  Place  of  Change  of  Date.  —  If  one  should  start  at 
any  point  on  the  earth  and  go  entirely  around  it  westward, 
the  number  of  times  the  sun  would  cross  his  meridian  would 
be  one  less  than  it  would  have  been  if  he  had  stayed  at  home. 
Since  it  would  be  very  inconvenient  for  him  to  use  fractional 
dates,  he  would  count  his  day  from  midnight  to  midnight, 
whatever  his  longitude,  and  correct  the  increasing  difference 
from  the  time  of  his  starting  point  by  arbitrarily  changing 
his  date  one  day  forward  at  some  point  in  his  journey.  That 
is,  he  would  omit  one  date  and  day  of  the  week  from  his 
reckoning.     On  the  other  hand,  if  he  were  going  around  the 


182     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  vi,  113 

earth  eastward,  he  would  give  two  days  the  same  date  and 
day  of  the  week.  The  change  is  usually  made  at  the  180th 
meridian  from  Greenwich.  This  is  a  particularly  fortunate 
selection,  for  the  180th  meridian  scarcely  passes  through  any 
land  surface  at  all,  and  then  only  small  islands.  One  can 
easily  see  how  troublesome  matters  would  be  if  the  change 
were  made  at  a  meridian  passing  through  a  thickly  popu- 
lated region,  say  the  meridian  of  Greenwich.     On  one  side 


IPS'    120°    135"    150'    165"    180'    165"     150"    135"    120°    IPS'    80'     78 
Fig.  66.  —  The  change-of-date  line. 

of  it  people  would  have  a  certain  day  and  date,  for  example, 
Monday,  December  24,  and  on  the  other  side  of  it  a  day 
later,  Tuesday,  December  25. 

The  place  of  actual  change  of  date  does  not  strictly  follow 
the  180th  meridian  from  Greenwich,  for  travelers,  going 
eastward  from  Europe,  lose  half  a  day,  while  those  going 
westward  from   Europe  and   America  arrive  in  the  same 


CH.  VI,  116']  TIME  183 

longitude  with  a  gain  of  half  a  day ;  hence  their  dates  differ 
by  one  day.     The  change-of-date  line  is  shown  in  Fig.  66. 

114.  The  Sidereal  Year.  —  The  sidereal  year  is  the  time 
required  for  the  sun  apparently  to  move  from  any  position 
with  respect  to  the  stars,  as  seen  from  the  earth,  around  to 
the  same  position  again.  Perhaps  it  is  better  to  say  that  it 
is  the  time  required  for  the  earth  to  make  a  complete  revolu- 
tion around  the  sun,  directions  from  the  sun  being  deter- 
mined by  the  positions  of  the  stars.  Its  length  in  mean 
solar  time  is  365  days,  6  hours,  9  minutes,  9.54  seconds,  or 
just  a  little  more  than  365.25  days. 

115.  The  Anomalistic  Year.  —  The  anomalistic  year  is 
the  time  required  for  the  earth  to  move  from  the  perihelion 
of  its  orbit  around  to  the  perihelion  again.  If  the  perihelion 
point  were  fixed,  this  period  would  equal  the  sidereal  year. 
But  the  attraction  of  the  other  planets  causes  the  perihelion 
point  to  move  forward  at  such  a  rate  that  it  completes  a 
revolution  in  about  108,000  years ;  and  the  consequence  is 
that  the  anomalistic  year  is  a  little  longer  than  the  sidereal 
year.  It  follows  from  the  period  of  its  revolution  that  the 
perihelion  point  advances  about  12"  annually.  Since  the 
earth  moves,  on  the  average,  about  a  degree  daily,  it  takes  it 
about  4  minutes  and  40  seconds  of  time  to  move  12".  The 
actual  length  of  the  anomalistic  year  in  mean  solar  time  is 
365  days,  6  hours,  13  minutes,  53.01  seconds. 

116.  The  Tropical  Year.  —  The  tropical  year  is  the  time 
required  for  the  sun  to  move  from  a  tropic  around  to  the 
same  tropic  again;  or,  better  for  practical  determination, 
from  an  equinox  to  the  same  equinox  again.  Since  the 
equinoxes  regress  about  50".2  annually,  the  tropical  year  is 
about  20  minutes  shorter  than'the  sidereal  year.  Its  actual 
length  in  mean  solar  time  is  365  days,  5  hours,  48  minutes, 
45.92  seconds. 

The  seasons  depend  upon  the  sun's  place  with  respect 
to  the  equinoxes.  Consequently,  if  the  seasons  are  always 
to  occur  at  the  same  time  according  to  the  calendar,  the 


184     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  vi,  116 

tropical  year  must  be  used.  This  is,  indeed,  the  year  in 
common  use  and,  unless  otherwise  specified,  the  term  year 
means  the  tropical  year. 

117.  The  Calendar.  —  In  very  ancient  times  the  calendar 
was  based  largely  on  the  motions  of  the  moon,  whose  phases 
determined  the  times  of  religious  ceremonies.  The  moon 
does  not  make  an  integral  number  of  revolutions  in  a  year, 
and  hence  it  was  occasionally  necessary  to  interpolate  a 
month  in  order  to  keep  the  year  in  harmony  with  the  seasons. 

The  week  was  another  division  of  time  used  in  antiquity. 
The  number  of  days  in  this  period  was  undoubtedly  based 
upon  the  number  of  moving  celestial  bodies  which  were  then 
known.  Thus,  Sunday  was  the  sun's  day;  Monday,  the 
moon's  day;  Tuesday,  Mars'  day;  Wednesday,  Mercury's 
day;  Thursday,  Jupiter's  day;  Friday,  Venus's  day;  and 
Saturday,  Saturn's  day.  The  names  of  the  days  of  the 
week,  when  traced  back  to  the  tongues  from  which  English 
has  been  derived,  show  that  these  were  their  origins. 

In  the  year  46  b.c.  the  Roman  calendar,  which  had 
fallen  into  a  state  of  great  confusion,  was  reformed  by 
Julius  Caesar  under  the  advice  of  an  Alexandrian  astronomer, 
Sosigenes.  The  new  system,  called  the  Julian  Calendar, 
was  entirely  independent  of  the  moon ;  in  it  there  were  3 
years  of  365  days  each  and  then  one  year,  the  leap  year,  of 
366  days.  This  mode  of  reckoning,  which  makes  the  aver- 
age year  consist  of  365.25  days,  was  put  into  effect  at  the 
beginning  of  the  year  45  b.c. 

It  is  seen  from  the  length  of  the  tropical  year,  which  was 
given  in  Art.  116,  that  this  system  of  calculation  involves  a 
small  error,  averaging  11  minutes  and  14  seconds  yearly. 
In  the  course  of  128  years  the  JuUan  Calendar  gets  one  day 
behind.  To  remedy  this  small  error,  in  1582,  Pope  Gregory 
XIII  introduced  a  sUght  change.  Ten  days  were  omitted 
from  that  year  by  making  October  15  follow  inunediately 
after  October  4,  and  it  was  decreed  that  3  leap  years  out  of 
every  4  centuries  should  henceforth  be  omitted.     This  again 


CH.  VI,  118]  TIME  185 

is  not  quite  exact,  for  the  Julian  Calendar  gets  behind  3 
days  in  3  X  128  =  384  years  instead  of  400  years ;  yet 
the  error  does  not  amount  to  a  day  until  after  more  than  3300 
years  have  elapsed. 

To  simplify  the  apphcation,  every  year  whose  date 
number  is  exactly  divisible  by  4  is  a  leap  year,  unless  it  is 
exactly  divisible  by  100.  Those  years  whose  date  numbers 
are  divisible  by  100  are  not  leap  years  unless  they  are  exactly 
divisible  by  400,  when  they  are  leap  years.  Of  course,  the 
error  which  still  remains  could  be  further  reduced  by  a  rule 
for  the  leap  years  when  the  date  number  is  exactly  divisible 
by  1000,  but  there  is  no  immediate  need  for  it. 

The  calendar  originated  and  introduced  by  Pope  Gregory 
XIII  in  1582,  and  known  as  the  Gregorian  Calendar,  is  now 
in  use  in  all  civilized  countries  except  Russia  and  Greece, 
although  it  was  not  adopted  in  England  until  1752.  At  that 
time  11  days  had  to  be  omitted  from  the  year,  causing  con- 
siderable disturbance,  for  many  people  imagined  they  were 
in  some  way  being  cheated  out  of  that  much  time.  The 
Julian  Calendar  is  now  13  days  behind  the  Gregorian  Calen- 
dar. The  Julian  Calendar  is  called  Old  Style  (O.S.),  and 
the  Gregorian,  New  Style  (N.S.). 

In  certain  astronomical  work,  such  as  the  discussion  of  the 
observations  of  variable  stars,  where  the  difference  in  time  of 
the  occurrence  of  phenomena  is  a  point  of  much  interest,  the 
Julian  Day  is  used.  The  Julian  Day  is  simply  the  number 
of  the  day  counting  forward  from  January  1,  4713  b.c.  This 
particular  date  from  which  to  count  time  was  chosen  because 
that  year  was  the  first  year  in  several  subsidiary  cycles, 
which  will  not  be  considered  here. 

118.  Finding  the  Day  of  the  Week  on  Any  Date.  —  An 
ordinary  year  of  365  days  consists  of  52  weeks  and  one  day, 
and  a  leap  year  consists  of  52  weeks  and  2  days.  Conse- 
quently, in  succeeding  years  the  same  date  falls  one  day 
later  in  the  week  except  when  a  twenty-ninth  of  February 
intervenes;    and  in  this  case  it  is  two  days  later.    These 


186     AN    INTRODUCTION   TO  ASTRONOMY     [ch.  vi,  118 

facts  give  the  basis  for  determining  the  day  of  the  week  on 
which  any  date  falls,  after  it  has  been  given  in  a  particular 
year. 

Consider  first  the  problem  of  finding  the  day  of  the  week 
on  which  January  1  falls.  In  the  year  1900  January  1  fell 
on  Monday.  To  fix  the  ideas,  consider  the  question  for 
1915.  If  every  year  had  been  an  ordinary  year,  January  1 
coming  one  day  later  in  the  week  in  each  succeeding  year, 
it  would  have  fallen,  in  1915,  15  days,  or  2  weeks  and  one 
day,  after  Monday;  that  is,  on  Tuesday.  But,  in  the 
meantime  there  were  3  leap  years,  namely,  1904,  1908, 
and  1912,  which  put  the  date  3  additional  days  forward  in 
the  week,  or  on  Friday.  Similarly,  it  is  seen  in  general 
that  the  rule  for  finding  the  day  of  the  week  on  which 
January  1  falls  in  any  year  of  the  present  centiu-y  is  to  take 
the  number  of  the  year  in  the  century  (15  in  the  example 
just  treated),  add  to  it  the  number  of  leap  years  which  have 
passed  (which  is  given  by  dividing  the  number  of  the  year 
by  4  and  neglecting  the  remainder),  divide  the  result  by 
7  to  eliminate  the  number  of  weeks  which  have  passed,  and 
finally,  count  forward  from  Monday  the  number  of  days 
given  by  the  remainder.  For  example,  in  1935  the  number 
of  the  year  is  35,  the  number  of  leap  years  is  8,  the  sum  of 
35  and  8  is  43,  and  43  divided  by  7  equals  6  with  the  remain- 
der of  1.  Hence,  in  1935,  January  1  will  be  one  day  later 
than  Monday ;  that  is,  it  will  fall  on  Tuesday. 

In  order  to  find  the  day  of  the  week  on  which  any  date  of 
any  year  falls,  find  first  the  day  of  the  week  on  which  Janu- 
ary 1  falls ;  then  take  the  day  of  the  year,  which  can  be 
obtained  by  adding  the  number  of  days  in  the  year  up  to  the 
date  in  question,  and  divide  this  by  7 ;  the  remainder  is  the 
number  of  days  that  must  be  added  to  that  on  which  Janu- 
ary 1  falls  in  order  to  obtain  the  day  of  the  week.  For 
example,  consider  March  21,  1935.  It  has  been  found  that 
January  1  of  this  year  falls  on  Tuesday.  There  are  79  days 
from  January  1  to  March  21  in  ordinary  years.     If  79  is 


CH.  VI,  118]  TIME  187 

divided  by  7,  the  quotient  is  11  with  the  remainder  of  2. 
Consequently,  March  21,  1935,  falls  2  days  after  Tuesday, 
that  is,  on  Thursday. 

IX.    QUESTIONS 

1.  Give  three  examples  where  intervals  of  time  in  which  you 
have  had  many  and  varied  intellectual  experiences  now  seem  longer 
than  ordinary  intervals  of  the  same  length.  Have  you  had  any  con- 
tradictory experiences  ? 

2.  If  the  sky  were  always  "covered  with  clouds,  how  should  we 
measure  time  ?  ■ 

3.  What  is  your  sidereal  time  to-day  at  8  p.m.  ? 

4.  What  would  be  the  relations  of  solar  time  to  sidereal' time  if 
the  earth  rotated  in  the  opposite  direction? 

5.  What  is  the  length  of  a  sidereal  day  expressed  in  mean  solar 
time? 

6.  What  is  the  standard  time  of  a  place  whose  longitude  is  85° 
west  of  Greenwich  when  its  local  time  is  11  a.m.  ? 

7.  What  is  the  local  time  of  a  place  whose  longitude  is  112°  west 
of  Greenwich  when  its  standard  time  is  8  p.m.  ? 

8.  Suppose  a  person  were  able  to  travel  around  the  earth  in  2 
days ;  what  would  be  the  lengths  of  his  days  and  nights  if  he  traveled 
from  east  to  west  ?    From  west  to  east  ? 

9.  If  the  sidereal  year  were  in  ordinary  use,  how  long  before 
Christmas  would  occur  when  the  sun  is  at  the  vernal  equinox  ? 

10.   On  what  days  of  the  week  will  your  birthday  fall  for  the  next 
12  years  ? 


CHAPTER   VII 
THE   MOON 

119.   The  Moon's  apparent  Motion  among  the  Stars.  — 

The  apparent  motion  of  the  moon  can  be  determined  by 
observation  without  any  particular  reference  to  its  actual 
motion.  In  fact,  the  ancient  Greeks  observed  the  moon 
with  great  care  and  learned  most  of  the  important  pecul- 
iarities of  its  apparent  motion,  but  they  did  not  know  its 
distance  from  the  earth  and  had  no  accurate  ideas  of  the 
character  of  its  orbit.  The  natural  method  of  procedure  is 
first  to  find  what  the  appearances  are,  and  from  them  to 
infer  the  actual  facts. 

The  moon  has  a  diurnal  motion  westward  which  is  pro- 
duced, of  course,  by  the  eastward  rotation  of  the  earth. 
Every  one  is  familiar  with  the  fact  that  it  rises  in  the  east, 
goes  across  the  sky  westward,  and  sets  in  the  west.  Those 
who  have  observed  it  except  in  the  most  casual  way,  have 
noticed  that  it  rises  at  various  points  on  the  eastern  horizon, 
crosses  the  meridian  at  various  altitudes,  and  sets  at  various 
points  on  the  western  horizon.  They  have  also  noticed  that 
the  interval  between  its  successive  passages  across  the 
meridian  is  somewhat  more  than  24  hours. 

Observations  of  the  moon  for  two  or  three  hours  will  show 
that  it  is  moving  eastward  among  the  stars.  When  its  path 
is  carefully  traced  out  during  a  whole  revolution,  it  is  found 
that  its  apparent  orbit  is  a  great  circle  which  is  inclined  to 
the  ecliptic  at  an  angle  of  5°  9'.  The  point  at  which  the 
moon,  in  its  motion  eastward,  crosses  the  ecliptic  from  south 
to  north  is  called  the  ascending  node  of  its  orbit,  and  the 
point  where  it  crosses  the  ecliptic  from  north  to  south  is 
called  the  descending  node  of  its  orbit.     The  attraction  of  the 

188 


CH.  VII,  120]  THE   MOON  189 

sun  for  the  moon  causes  the  nodes  continually  to  regress  on 
the  ecliptic;  that  is,  in  successive  revolutions  the  moon 
crosses  the  ecliptic  farther  and  farther  to  the  west.  The 
period  of  revolution  of  the  Une  of  nodes  is  18.6  years. 

120.  The  Moon's  Synodical  and  Sidereal  Periods.  —  The 
synodical  period  of  the  moon  is  the  time  required  for  it  to 
move  from  any  apparent  position  with  respect  to  the  sun 
back  to  the  same  position  again.  The  most  accurate  means 
of  determining  this  period  is  by  comparing  ancient  and 
modern  echpses  of  the  sun ;  for,  at  the  time  of  a  solar  eclipse, 
the  moon  is  exactly  between  the  earth  and  the  sun.  The 
advantages  of  this  method  are  that,  in  the  first  place,  at  the 
epochs  used  the  exact  positions  of  the  moon  with  respect  to 
the  sun  are  known ;  and,  in  the  second  place,  in  a  long  inter- 
val during  which  the  moon  has  made  hundreds  or  even 
thousands  of  revolutions  around  the  earth,  the  errors  in  the 
determinations  of  the  exact  times  of  the  eclipses  are  rela- 
tively unimportant  because  they  are  divided  by  the  number 
of  revolutions  the  moon  has  performed.  It  has  been  found 
in  this  way  that  the  synodical  period  of  the  moon  is  29  days, 
12  hours,  44  minutes,  and  2.8  seconds ;  or  29.530588  days, 
with  an  uncertainty  of  less  than  one  tenth  of  a  second. 

The  sidereal  period  of  the  moon  is  the  time  required  for 
it  to  move  from  any  apparent  position  with  respect  to  the 
stars  back  to  the  same  position  again.  This  period  can  be 
found  by  direct  observations ;  or,  it  can  be  computed  from 
the  synodical  period  and  the  period  of  the  earth's  revolution 
around  the  sun.  Let  S  represent  the  moon's  synodical 
period,  M  its  sidereal  period,  and  E  the  period  of  the  earth's 
revolution  around  the  sun,  all  expressed  in  the  same  units  as, 
for  example,  days.  Then  1/M  is  the  fraction  of  a  revolution 
that  the  moon  moves  eastward  in  one  day,  1/E  is  the  fraction 
of  a  revolution  that  the  sun  moves  eastward  in  one  day, 
and  1/M-  1/  E  is,  therefore,  the  fraction  of  a  revolution  that 
the  moon  gains  on  the  sun  in  its  eastward  motion  in  one  day. 
Since  the  moon  gains  one  complete  revolution  on  the  sun  in 


190    AN   INTRODUCTION   TO  ASTRONOMY    [ch.  vii,  120 

S  days,  1/S  is  also  the  fraction  of  a  revolution  the  moon 
gains  on  the  sun  in  one  day.     Hence  it  follows  that 

1_^   1 1 

S      M     E' 

from  which  M  can  be  computed  when  S  said  E  are  known. 

It  is  easy  to  see  that  the  synodical  period  is  longer  than 
the  sidereal.  Suppose  the  sun,  moon,  and  certain  stars  are 
at  a  given  instant  in  the  same  straight  line  as  seen  from  the 
earth.  After  a  certain  number  of  day^  the  moon  will  have 
made  a  sidereal  revolution  and  the  sun  will  have  moved  east- 
ward among  the  stars  a  certain  number  of  degrees.  Since 
additional  time  is  required  for  the  moon  to  overtake  it,  the 
synodical  period  is  longer  than  the  sidereal. 

It  has  been  found  by  direct  observations,  and  also  by  the 
equation  above,  that  the  moon's  sidereal  period  is  27  days, 
7  hours,  43  minutes,  and  11.5  sfeconds,  or  27.32166  days. 
When  the  period  of  the  moon  is  referred  to,  the  sidereal 
period  is  meant  unless  otherwise  stated. 

The  periods  which  have  been  given  are  averages,  for  the 
moon  departs  somewhat  from  its  elliptical  orbit,  primarily 
because  of  the  attraction  of  the  sun,  and  to  a  lesser  extent 
because  of  the  oblateness  of  the  earth  and  the  attractions  of 
the  planets.  The  variations  from  the  average  are  sometimes 
quite  appreciable,  for  the  perturbations,  as  they  are  called, 
may  cause  the  moon  to  depart  from  its  undisturbed  orbit 
about  1°.5,  and  may  cause  its  period  of  revolution  to  vary  by 
as  much  as  2  hours. 

121.  The  Phases  of  the  Moon.  —  The  moon  shines  en- 
tirely by  reflected  sunlight,  and  consequently  its  appearance 
as  seen  from  the  earth  depends  upon  its  position  relative  to 
the  sun.  Figure  67  shows  eight  positions  of  the  moon  in  its 
orbit  with  the  sun's  rays  coming  from  the  right  in  lines  which 
are  essentially  parallel  because  of  the  great  distance  of  the 
sun.  The  left-hand  side  of  the  earth  is  the  night  side,  and 
similarly  the  left  side  of  the  moon  is  the  dark  side. 


'    (' 

nd*    ,            1 

A       ■                 1 

\      ^ 

\       ' 

/KO»     - 

survi  pArs     1 

7   *        ' 

/    . 

^    ,            \ 

^   9>   / 

CH.  vii,  121]  THE   MOON  191 

The  small  circles  whose  centers  are  on  the  large  circle 
around  the  earth  as  a  center  show  the  illuminated  and  un- 
illuminated  parts  of  the  moon  as  they  actually  are;  the 
accompanying  small  circles  Just  outside  of  the  large  circle 
show  the  moon  as 
it  is  seen  from  the 
earth.  For  example, 
when  the  moon  is 
at  Ml  between  the 
earth  and  sun,  its 
dark  side  is  toward 
the  earth.  In  this 
position  it  is  said  to 
be  in  conjunction', 
and  the  phase  is  new. 

At     M      hnlf    of    thp      Fig.  67.  —  Explanation  of  the  moon's  phases. 

illuminated  part  of  the  moon  can  be  seen  from  the  earth,  and 
it  is"  in  the  first  quarter.  In  this  position  the  moon  is  said 
to  be  in  quadrature.  Between  the  new  moon  and  the  first 
quarter  the  illuminated  part  of  the  moon  as  seen  from  the 
earth  is  of  crescent  shape,  and  its  convex  side  is  turned 
toward  the  sun. 

When  the  moon  is  at  M3  the  illuminated  side  is  toward  the 
earth.  It  is  then  in  opposition,  and  the  phase  is  full.  If  an 
observer  were  at  the  sunset  point  on  the  earth,  the  sun 
would  be  setting  in  the  west  and  the  full  moon  would  be 
rising  in  the  east.  At  M4  the  moon  is  again  in  quadrature, 
and  the  phase  is  third  quarter. 

To  summarize :  The  moon  is  new  when  it  has  the  same 
right  ascension  as  the  sun ;  it  is  at  the  first  quarter  when 
its  right  ascension  is  6  hours  greater  than  that  of  the  sun ; 
it  is  full  when  its  right  ascension  is  12  hours  greater  than 
that  of  the  sun ;  and  it  is  at  the  third  quarter  when  its  right 
ascension  is  18  hours  greater  than  that  of  the  sun. 

It  is  observed  from  the  diagram  that  the  earth  would 
havt!  phases  if  seen  from  the  moon.     When  the  moon  is 


192    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  121 


new,  as  seen  frona  the  earth,  the  earth  would  be  full  as  seen 
from  the  moon.     The  phases  of  the  earth  corresponding  to 

every  other  position  of 
the  moon  can  be  inferred 
from  the  diagram.  The 
phases  of  the  moon  and 
earth  are  supplementary; 
that  is,  the  illuminated 
portion  of  the  moon  as 
seen  from  the  earth  plus 
the  illuminated  portion  of 
the  earth  as  seen  from  the 
moon  always  equals  180°. 
When  the  moon  is  nearly 
new,  and,  consequently, 
the  earth  nearly  full  as 
seen  from  the  moon,  the 
dark  side  of  the  moon  is 
somewhat  illuminated  by  simlight  reflected  from  the  earth, 
as  is  shown  in  Fig.  68. 

122.  The  diurnal  Circles  of  the  Moon.  —  Suppose  first 
that  the  moon  moves  along  the  ecliptic  and  consider  its 
diurnal   circles.     Since  they   are   parallel   to   the   celestial 


Fig.  68.  —  The  moon  partially  illu- 
minated by  light  reflected  from  the 
earth.  Photographed  by  Barnard  at 
the  Yerkes  Observatory. 


Fig.  69.  —  The  equator  and  ecliptic. 

equator  (if  the  motion  of  the  moon  in  declination  between 
rising  and  setting  is  neglected),  it  is  sufficient,  in  view  of  the 
discussion  of  the  sun's  diurnal  circles  (Art.  58),  to  give  the 
places  where  the  moon  crosses  the  meridian.  Let  VAV, 
Fig.  69,  represent  the  celestial  equator  spread  out  on  a  plane, 
and  VSAWV  the  ecliptic.  Suppose,  for  example,  that  the 
time  of  the  year  is  March  21.     Then  the  sun  is  at  V.    If 


CH.  VII,  122] 


THE   MOON 


193 


the  moon  is  new,  it  is  also  at  V,  because  at  this  phase  it  has 
the  same  right  ascension  as  the  sun.  Since  V  is  on  the  celes- 
tial equator,  the  moon  crosses  the  meridian  at  an  altitude 
equal  to  90°  minus  the  latitude  of  the  observer.  In  this 
case  it  rises  in  the  east  and  sets  in  the  west.  But  if  the  moon 
is  at  first  quarter  on  March  21,  it  is  at  S,  because  at  this 
phase  it  is  6  hours 'east  of  the  sun.  It  is  then  23°. 5  north 
of  the  equator,  and,  consequently,  it  crosses  the  meridian 
23°.5  above  the  equator.  In  this  case  it  rises  north  of  east 
and  sets  north  of  west.  If  the  moon  is  full,  it  is  at  4,  and 
if  it  is  in  the  third  quarter,  it  is  at  W.  In  the  former  case  it 
is  on  the  equator  and  in  the  latter  23°. 5  south  of  it. 

Suppose  the  sun  is  at  the  summer,  solstice,  S.  Then  it 
rises  in  the  northeast,  crosses  the  meridian  23°. 5  north  of 
the  equator,  and  sets  in  the  northwest.  At  the  same  time 
the  full  moon  is  at  W,  it  rises  in  the  southeast,  crosses  the 
meridian  23°.5  south  of  the  equator,  and  sets  in  the  south- 
west. That  is,  when  sunshine  is  most  abundant,  the  light 
from  the  full  moon  is  the  least.  On  the  other  hand,  when 
the  sun  is  at  the  winter  solstice  W,  the  full  moon  is  at  S 
and  gives  the  most  light.  The  other  positions  of  the  sun 
and  moon  can  be  treated  similarly. 

Suppose  the  ascending  node  of  the  moon's  orbit  is  at  the 
vernal  equinox  (Fig.  70),  and  consider  the_  altitude  at  which 


;? 

..^ft^ 

~- 

r^^-^^S^!^  ' 

E0UArOR 

1/ 

-AJ* 

'' 

^^**->.^^   , 

^^^^<^ 

^ 

-^* 

-~  -  -  1 - 

Fig.  70.  —  Ascending  node  of  the  moon's  orbit  at  the  vernal  equinox. 


the  moon  crosses  the  meridian  when  full  at  the  time  of  the 
winter  solstice.  The  sun  is  at  W  and  the  full  moon  is  in  its 
orbit  5°  9'  north  of  &.  If  the  latitude  of  the  observer  is  40°, 
the  moon  then  crosses  his  meridian  at  an  altitude  of  50°  + 
23°.5  -f-  5°  =  78°.5.  That  is,  under  these  circumstances  the 
o 


194    AN  INTRODUCTION   TO  ASTRONOMY    [ch.  vii,  122 

full  moon  crosses  the  meridian  higher  in  the  winter  time 
than  it  would  if  its  orbit  were  coincident  with  the  ecliptic. 
On  the  other  hand,  in  the  summer  time,  when  the  sun  is  at 
S  and  the  full  moon  is  at  W,  the  moon  crosses  the  equator 
farther  south  than  it  would  if  it  were  on  the  ecliptic.  Under 
these  circumstances  there  is  more  moonlight  in  the  winter 
and  less  in  the  summer  than  there  would  be  if  the  moon 
were  always  on  the  ecliptic. 

Now  suppose  the  descending  node  is  at  V  and  the  ascend- 
ing node  is  Sbt  A,  Fig.  71.     For  this  position  of  its  orbit  the 


Fig.  71.  —  Ascending  node  of  the  moon's  orbit  at  the  autumnal  equinox. 

moon  crosses  the  meridian  lower  in  the  winter  than  it  would 
if  it  moved  along  the  ecliptic.  The  opposite  is  true  when 
the  sun  is  at  S  in  the  summer.  Of  course,  the  ascending 
node  of  the  moon's  orbit  might  be  at  any  other  point  on 
the  ecliptic. 

It  is  clear  from  this  discussion  that  when  the  sun  is  on 
the  part  of  the  ecliptic  south  of  the  equator,  the  fuU  moon 
is  near  the  part  of  the  ecliptic  which  is  north  of  the  equator, 
and  vice  versa.  Therefore,  when  there  is  least  sunhght  there 
is  most  moonlight,  and  there  is  the  greatest  amount  of  moon- 
light when  the  moon's  ascending  node  is  at  the  vernal 
equinox.  When  it  is  continuous  night  at  a  pole  of  the  earth, 
the  gloom  is  partly  dispelled  by  the  moon  which  is  above  the 
horizon  that  half  of  the  month  in  which  it  passes  from  its 
first  to  its  third  quarter. 

123.  The  Distance  of  the  Moon.  —  One  method  of  deter- 
mining the  distance  of  the  moon  is  by  observing  the  differ- 
ence in  its  directions  as  seen  from  two  points  on  the  earth's 
surface,  as  Oi  and  O2  in  Fig.  72.  Suppose,  for  simplicity, 
that  Oi  and  O2  are  on  the  same  meridian,  and  that  the  moon 


CH.  vn,  123]  THE   MOON  195 

is  in  the  plane  of  that  meridian.  The  observer  at  d  finds 
that  the  moon  is  the  angular  distance  ZiOiM  south  of  his 
zenith ;  and  the  observer  at  O2  finds  that  it  is  the  angular 
distance  Z2O2M  north  of  his  zenith.  Since  the  two  observ- 
ers know  their  latitudes,  they  know  the  angle  O1EO2,  and 
consequently,  the  angles  EO1O2  and  EO^Oi.  By  subtract- 
ing ZiOiM  plus  EO1O2  and  Z2O2M  plus  EO2O1  from  180°, 
the  angles  MO1O2  and  MO2O1  are  obtained.  Since  the  size 
of  the  earth  is  known,  the  distance  O1O2  can  be  found.  Then, 
in  the  triangle  O1MO2  two  angles  and  the  included  side  are 
known,  and  all  the  other  parts  of  the  triangle  can  be  com- 
puted by  trigonometry.     Suppose  OiM  has  been  found; 

.3« 


£■•'-', 

----Vv 

>v^ 

^■^T 

X 

f" 
Fig. 

72.- 

/OI-- 


■  Measuring  the  distance  to  the  moon. 


then,  in  the  triangle  EO\M  two  sides  and  the  included  angle 
are  known,  and  the  distance  EM  can  be  computed.  In 
general,  the  relations  and  observations  wiU  not  be  so  simple 
as  those  assimied  here,  but  in  no  case  are  serious  mathe- 
matical or  observational  difficulties  encountered.  It  is  to 
be  noted  that  the  result  obtained  is  not  guesswork,  but 
that  it  is  based  on  measurements,  and  that  it  is  in  reaHty 
given  by  measurements  in  the  same  sense  that  a  distance 
on  the  surface  of  the  earth  may  be  obtained  by  measure- 
hient.  The  percentage  of  error  in  the  determination  of  the 
moon's  distance  is  actually  much  less  than  that  in  most  of 
the  ordinary  distances  on  the  surface  of  the  earth. 


196    AN   INTRODUCTION   TO  ASTRONOMY    [ch.  vii,  123 

The  mean  distance  from  the  center  of  the  earth  to  the 
center  of  the  moon  has  been  found  to  be  238,862  miles,  and 
the  circumference  of  its  orbit  is  therefore  1,500,818  miles. 
On  dividing  the  circumference  by  the  moon's  sidereal  period 
expressed  in  hours,  it  is  found  that  its  orbital  velocity  aver- 
ages 2288.8  miles  per  hour,  or  about  3357  feet  per  second. 

A  body  at  the  surface  of  the  earth  falls  about  16  feet  the 
first  second ;  at  the  distance  of  the  moon,  which  is  approxi- 
mately 60  times  the  radius  of  the  earth,  it  would,  therefore, 
fall  16  ^  60^  =  0.0044  feet,  because  the  earth's  attraction 
varies  inversely  as  the  square  of  the  distance  from  its  center. 
Therefore,  in  going  3357  feet,  or  nearly  two  thirds  of  a  mile, 
the  moon  deviates  from  a  straight-line  path  only  about  -^ 
of  an  inch. 

124.  The  Dimensions  of  the  Moon.  —  The  mean  apparent 
diameter  of  the  moon  is  31'  5".2.  Since  its  distance  is 
known,  its  actual  diameter  can  be  computed.  It  is  found 
that  the  distance  straight  through  the  moon  is  2160  miles, 
or  a  httle  greater  than  one  fourth  the  diameter  of  the  earth. 
Since  the  surfaces  of  spheres  are  to  each  other  as  the  squares 
of  their  diameters,  it  is  found  that  the  surface  area  of  the 
earth  is  13.4  times  that  of  the  moon ;  and  since  the  volumes 
of  spheres  are  to  each  other  as  the  cubes  of  their  diameters, 
it  is  found  that  the  volume  of  the  earth  is  49.3  times  that 
of  the  moon. 

It  has  been  stated  that  the  mean  apparent  diameter  of 
the  moon  is  31'  5".2.  The  apparent  diameter  of  the  moon 
varies  both  because  its  distance  from  the  center  of  the  earth 
varies,  and  also  because  when  the  moon  is  on  the  observer's 
meridian,  he  is  nearly  4000  miles  nearer  to  it  than  when 
it  is  on  his  horizon.  In  the  observations  of  other  celestial 
objects  the  small  distance  of  4000  miles  makes  no  appre- 
ciable difference  in  their  appearance;  but,  since  the  dis- 
tance from  the  earth  to  the  moon  is,  in  round  numbers,  only 
240,000  miles,  the  radius  of  the  earth  is  ^  of  the  whole 
amount. 


CH.  VII,  126]  THE   MOON  197 

In  spite  of  the  fact  that  the  moon  is  nearer  the  observer 
when  it  is  on  his  meridian  than  when  it  is  on  his  horizon, 
every  one  has  noticed  that  it  appears  largest  when  near 
the  horizon  and  smallest  when  near  the  meridian.  The 
reason  that  the  moon  appears  to  us  to  be  larger  when  it  is 
near  the  horizon  is  that  then  intervening  objects  give  us  the 
impression  that  it  is  very  distant,  and  this  influences  our 
unconscious  estimate  of  its  size. 

125.  The  Moon's  Orbit  with  Respect  to  the  Earth.  — 
The  moon's  distance  from  the  earth  varies  from  about 
225,746  miles  to  251,978  miles,  causing  a  .corresponding 
variation  in  its  apparent  diameter.  Its  orbit  is  an  eUipse, 
having  an  eccentricity  of  0.0549,  except  for  slight  deviations 
due  to  the  attractions  of  the  sun,  planets,  and  the  equatorial 
bulge  of  the  earth.  The  moon  moves  around  the  earth, 
which  is  at  one  of  the  foci  of  its  elliptical  orbit,  in  such  a 
manner  that  the  hne  joining  it  to  the  earth  sweeps  over 
equal  areas  in  equal  intervals  of  time.  This  statement  re- 
quires a  slight  correction  because  of  the  perturbations  pro- 
duced by  the  attractions  of  the  sun  and  planets.  The 
point  in  the  moon's  orbit  which  is  nearest  the  earth  is  called 
its  perigee,  and  the  farthest  point  is  called  its  apogee. 

126.  The  Moon's  Orbit  with  Respect  to  the  Sun.  —  The 
distance  from  the  earth  to  the  sun  is  about  400  times  that 
from  the  earth  to  the  moon.  Consequently,  the  oscillations 
of  the  moon  back  and  forth  across  the  earth's  orbit  as  the 
two  bodies  pursue  their  motion  around  the  sun  are  so  small 
that  they  can  hardly  be  represented  to  scale  in  a  diagram. 
As  a  consequence  of  the  relative  nearness  of  the  moon  and 
its  comparatively  long  period,  its  orbit  is  always  concave 
towa,rd  the  sun.  If  the  orbit  of  the  moon  were  at'  any  time 
convex  toward  the  sun,  it  would  be  when  it  is  moving  from 
a  position  between  the  earth  and  sun  to  opposition,  that 
is,  from  A  to  B,  Fig.  73.  It  takes  14  days  for  the  moon 
to  move  from  the  former  position  to  the  latter,  and  during 
this  time  its  distance  from  the  sun  increases  by  about  480,000 


198     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  vii,  126 

miles;  but,  in  the  meantime,  the  earth  moves  forward 
about  14°  in  its  orbit  from  P  to  Q,  and  it,  therefore,  is  drawn 
by  the  sun  away  from  the  straight  Une  PT  in  which  it  was 
originally  moving  by  a  distance  of  about  3,000,000  miles, 


A 
Fig.  73.  —  The  orbit  of  the  moon  is  concave  to  the  sun. 

That  is,  in  the  14  days  the  moon  actually  moves  in  toward 
the  sun  away  from. the  original  line  of  the  earth's  motion 
3,000,000  -  480,000  =  2,520,000  miles,  and  its  orbit,  which 
is  represented  by  the  broken  line,  is,  therefore,  concave  toward 
the  sun  at  every  point. 

As  a  matter  of  fact,  it  is  the  center  of  gravity  of  the  earth 
and  moon  which  describes  what  is  called  the  earth's  ellip- 
tical orbit  around  the  sun,  and  the  earth  and  moon  both 
describe  ellipses  around  this  point  as  it  moves  on  in  its  ellip-; 
tical.path  around  the  sun.  Since  the  earth's  mass  is  very 
large  compared  to  that  of  the  moon,  as  will  be  seen  in  Art. 
127,  the  center  of  the  earth  is  always  very  near  the  center 
of  gravity  of  the  two  bodies. 

127.  The  Mass  of  the  Moon.  —  Although  the  moon  is 
comparatively  near  the  earth,  its  mass  cannot  be  obtained  so 
easily  as  that  of  many  other  objects  farther  away. 

One  of  the  best  methods  of  finding  the  mass  of  the  moon 
depends  upon  the  fact  that  the  center  of  gravity  of  the 
earth  and  moon  describes  an  elliptical  orbit  around  the  sun 
in  accordance  with  the  law  of  areas.  Sometimes  the  earth 
is  ahead  of  the  center  of  gravity,  and  at  other  times  behind 
it.  When  the  earth  is  ahead  of  the  center  of  gravity  the 
sun  will  be  seen  behind  the  position  it  would  apparently 
occupy  if  it  were  not  for  the  moon.     On  the  other  hand, 


CH.  vii,  127]  THE   MOON  199 

when  the  earth  is  behind  the  center  of  gravity,  the  sun  will 
be  displaced  correspondingly  ahead  of  the  position  it  would 
otherwise  apparently  occupy.  That  is,  the  sun's  apparent 
motion  eastward  among  the  stars  is  not  strictly  in  accord- 
ance with  the  law  of  areas,  for  it  sometimes  is  a  little  ahead 
of,  and  at  others  a  little  behind,  the  position  it  would  have 
except  for  the  moon.  From  very  delicate  observations  it 
has  been  found  that  the  sun  is  displaced  in  this  way  about 
6".4.  Since  the  distance  of  the  sun  is  known,  the  amount 
of  displacement  of  the  earth  in  miles  necessary  to  produce 
this  apparent  displacement  of  the  sun  can  be  computed. 
It  has  been  found  in  this  way  that  the  distance  of  the  center 
of  gravity  of  the  earth  and  moon  from  the  center  of  the  earth 
is  2886  miles. 

Now  consider  the  problem  of  finding  the  ratio  of  the  mass 
of  the  earth  to  that  of  the  moon.  In  Fig.  74  let  E  represent 
the  earth,  C  the  center 
of  gravity  of  the  earth 

and   moon,   and  M  the    [       £_?_?] t-' Q 

moon.     Let  the  distance 
EC  be  represented  by  x, 

J     .1         J-  i.  EiTi/T      Fig.  74.  —  Center  of  gravity  of  the  earth 

and  the    distance   EM,  and  moon. 

which  is  238,862  miles, 

by  r.     Since  the  mass  of  the  earth  multiplied  by  the  distance 

of  its  center  from  the  center  of  gravity  of  the  earth  and  moon 

equals  the  mass  of  the  moon  multipHed  by  its  distance  from  ■ 

the  center  of  gravity  of  the  earth  and  moon,  it  follows  that 

X  X  E  =  (r-x)  M. 

Since  x  =  2886  miles  and  r  =  238,862  miles,  it  is  found 

that 

E  =  81.8  M. 

In  round  numbers  the  mass  of  the  earth  is  80  times  that  of 
the  moon. 

Since  the  orbit  of  the  moon  is  inclined  5°  9'  to  the  plane 
of  the  ecliptic,  the  earth  is  sometimes  above  and  sometimes 


200    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  127 

below  this  plane.  This  causes  an  apparent  displacement  of 
the  sun  from  the  ecliptic  in  the  opposite  direction.  From 
the  amount  of  the  apparent  displacement  of  the  sun  in 
latitude,  as  determined  by  observations,  and  from  the  in- 
cUnation  of  the  moon's  orbit  and  the  distance  of  the  sun, 
it  is  possible  to  compute,  just  as  from  the  sun's  apparent 
displacement  in  longitude,  the  mass  of  the  moon  relative  to 
that  of  the  earth. 

128.  The  Rotation  of  the  Moon.  —  The  moon  always 
presents  the  same  side  toward  the  earth,  and  therefore,  as 
seen  from  some  point  other  than  the  earth  or  moon,  it  ro- 
tates on  its  axis  once  in  a  sidereal  month.  For,  in  Fig.  67, 
when  the  moon  is  at  Mi  a  certain  part  is  on  the  left  toward 
the  earth,  but  when  it  has  moved  to  Ma  the  same  side  is  on 
the  right  stiU  toward  the  earth.  Its  direction  of  rotation 
is  the  same  as  that  of  its  revolution,  or  from  west  to  east. 
The  plane  of  its  equator  is  inclined  about  1°  32'  to  the  plane 
of  the  ecliptic,  and  the  two  planes  always  intersect  in  the 
line  of  nodes  of  the  moon's  orbit. 

It  follows  from  what  has  been  stated  that  the  moon's 
sidereal  day  is  the  same  as  its  sidereal-  month,  or  27.32166 
mean  solar  days.  Its  solar  day  is  of  the  same  length  as  its 
synodical  month,  or  29.530588  mean  solar  days,  because  its 
synodical  month  is  defined  by  its  position  with  respect  to 
the  earth  and  sun.  Other  things  being  equal,  the  tempera- 
ture changes  from  day  to  night  on  the  moon  would  be  much 
greater  than  on  the  earth  because  its  period  of  rotation  is  so 
much  longer ;  but  the  seasonal  changes  would  be  very  slight 
because  of  the  small  inclination  of  the  plane  of  its  equator 
to  the  plane  of  its  orbit. 

It  is  a  most  remarkable  fact  that  the  moon  rotates  at 
precisely  such  a  rate  that  it  always  keeps  the  same  face 
toward  the  earth.  It  is  infinitely  improbable  that  it  was 
started  exactly  in  this  way ;  and,  if  it  were  not  so  started, 
there  must  have  been  forces  at  work  which  have  brought 
about  this  peculiar  relationship.     It  has  been  suggested  that 


CH.  VII,  129]  THE   MOON  201 

the  explanation  lies  in  the  tidal  reaction  between  the  earth 
and  moon.  Since  the  moon  raises  tides  on  the  earth,  it  is 
obvious  that  the  earth  also  raises  tides  on  the  moon  unless 
it  is, absolutely  rigid.  Since  the  mass  of  the  earth  is  more 
than  80  times  that  of  the  moon,  the  tides  generated  by  the 
earth  on  the  moon,  other  things  being  equal,  would  be  much 
greater  than  those  generated  by  the  moon  on  the  earth.  If 
a  body  is  rotating  faster  than  it  revolves,  and  in  the  same 
direction,  one  of  the  effects  of  the  tides  is  to  slow  up  its 
rotation  and  to  tend  to  bring  the  periods  of  rotation  and 
revolution  to  an  equahty.  It  has  been  generally  beheved 
that  the  tides  raised  by  the  earth  on  the  moon  during  mil- 
lions of  years,  part  of  which  time  it  may  have  been  in  a 
plastic  state,  have  brought  about  the  condition  which  now 
exists.  There  are,  however,  serious  difficulties  with  this 
explanation  (Art.  265),  and  it  seems  probable  that  the  earth 
and  moon  are  connected  by  forces  not  yet  understood. 

129.  The  Librations  of  the  Moon.  —  The  statement  that 
the  moon  always  has  the  same  side  toward  the  earth  is  not 
true  in  the  strictest  sense.  It  would  be  true  if  the  planes 
of  its  orbit  and  of  its  equator  were  the  same,  and  if  it  moved 
at  a  perfectly  uniform  angular  velocity  in  its  orbit. 

The  inclination  of  the  moon's  orbit  to  the  ecliptic  averages 
about  5°  9',  and  the  inclination  of  the  moon's  equator  to 
the  echptic  is  about  1°  32'.  The  three  planes  are  so  related 
that  the  inclination  of  the  moon's  equator  to  the  plane  of 
its  orbit  is  5°  9'  +  1°  32'  =  6°  41'.  The  sun  shines  alter- 
nately over  the  two  poles  of  the  earth  because  of  the  incli- 
nation of  the  plane  of  the  equator  to  the  plane  of  the  ecliptic. 
In  a  similar  manner,  if  the  earth  were  a  luminous  body  it 
would  shine  6°  41'  over  the  moon's  poles.  Instead  of  shin- 
ing on  them  (except  by  reflected  hght),  the  tilting  of  the 
moon's  axis  of  rotation  enables  us  to  see  6°  41'  over  the  poles. 
This  is  the  libration  in  latitude. 

The  moon  rotates  at  a  uniform  rate,  —  at  least  the  depar- 
tures from  a  uniform  rate   are   absolutely  insensible.     It 


202    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  129 

would  take  inconceivably  great  forces  to  make  perceptible 
short  changes  in  its  rate  of  rotation.  On  the  other  hand, 
the  moon  revolves  around  the  earth  at  a  non-uniform  rate, 
for  it  moves  in  such  a  way  that  the  law  of  areas  is  fulfilled. 
Consider  the  moon  starting  from  the  perigee.  It  takes 
about  6.5  days,  or  considerably  less  than  one  quarter  of  its 
period,  for  the  moon  to  revolve  through  90° ;  and,  therefore, 
the  angle  of  rotation  is  considerably  less  than  90°.  The 
result  is  that  the  part  of  the  moon  on  the  side  toward  the 
perigee,  that  is,  the  western  edge,  is  brought  partially  into 
view.  On  the  opposite  side  of  the  orbit,  the  eastern  edge  of 
the  moon  is  brought  partially  into  view.  This  is  the  libra- 
tion  in  longitude. 

In  addition  to  this,  the  moon  is  not  viewed  from  the  earth's 
center.  When  it  is  on  the  horizon,  the  line  from  the  ob- 
server to  the  moon  makes  an  angle  of  nearly  1°  (the  parallax 
of  the  moon)  with  that  from  the  earth's  center  to  the  moon. 
This  enables  the  observer  to  see  nearly  1°  farther  around  its 
side  than  he  could  if  it  were  on  his  meridian. 

The  result  of  the  moon's  librations  is  that  there  is  only 
41  per  cent  of  its  surface  which  is  never  seen,  while  41  per 
cent  is  always  in  sight,  and  18  per  cent  of  it  is  sometimes 
visible  and  sometimes  invisible. 

130.  The  Density  and  Surface  Gravity  of  the  Moon.  — 
The  volume  of  the  earth  is  about  50  times  that  of  the  moon 
and  its  mass  is  81.8  times  that  of  the  moon.  Therefore  the 
density  of  the  moon  is  somewhat  less  than  that  of  the  earth. 
It  is  found  from  the  relative  volumes  and  masses  of  the  earth 
and  moon  that  the  density  of  the  moon  on  the  water  stand- 
ard is  about  3.4. 

If  the  radius  of  the  moon  were  the  same  as  that  of  the 
earth,  gravity  at  its  surface  would  be  less  than  ^  that  at 
the  surface  of  the  earth ;  but  the  small  radius  of  the  moon 
tends  to  increase  the  attraction  at  its  surface.  If  its  mass 
were  the  same  as  that  of  the  earth,  its  surface  gravity  would 
be  nearly  16  times  that  of  the  earth.     On  taking  the  two 


CH.  VII,  131]  THE  MOON  203 

factors  together,  it  is  found  that  the  surface  gravity  of  the 
moon  is  about  i  that  of  the  earth.  That  is,  a  body  on 
the  earth  weighs  by  spring  balances  about  6  times  as  much 
as  it  would  weigh  on  the  moon.  i 

If  a  body  were  thrown  up  from  the  surface  of  the  moon 
with  a  given  velocity,  it  would  ascend  6  times  as  high  as  it 
would  if  thrown  up  from  the  surface  of  the  earth  with  the 
same  velocity.  Perhaps  this  is  the  reason  why  the  forces 
to  which  both  the  earth  and  moon  have  been  subjected  have 
produced  relatively  higher  elevations  on  the  moon  than  on 
the  earth.  Also  it  would  be  possible  for  mountains  of  a 
given  material  to  be  6  times  as  high  on  the  moon  as  on  the 
earth  before  the  rock  of  which  they  are  composed  would  be 
crushed  at  the  bottom. 

131.  The  Question  of  the  Moon's  Atmosphere.  —  The 
moon  has  no  atmosphere,  or  at  the  most,  an  excessively  rare 
one.  Its  absence  is  proved  by  the  fact  that,  at  the  time  of 
an  eclipse  of  the  sun,  the  moon's  limb  is  perfectly  dark  and 
sharp,  with  no  apparent  distortion  of  the  sun  due  to  refrac- 
tion. Similarly,  when  a  star  is  occulted  by  the  moon,  it 
disappears  suddenly  and  not  somewhat  gradually  as  it 
would  if  its  light  were  being  more  and  more  extinguished 
by  an  atmosphere. 

Besides  this,  if  the  moon  had  an  atmosphere,  its  refraction 
would  keep  a  star  visible  for  a  little  time  after  it  had  been 
occulted,  just  as  the  earth's  atmosphere  keeps  the  sun 
visible  about  2  minutes  after  it  has  actually  set.  In  a  simi- 
lar way,  the  star  would  become  visible  a  short  time  before 
the  moon  had  passed  out  of  line  with  it.  The  whole  effect 
would  be  to  make  the  time  of  occultation  shorter  than  it 
would  be  if  there  were  no  atmosphere. 

If  the  moon  had  an  atmosphere  of  any  considerable 
extent,  there  would  be  the  effects  of  erosion  on  its  surface  ; 
but  so  far  as  can  be  determined,  there  is  no  evidence  of  such 
action.  Its  surface  consists  of  a  barren  waste,  and  it  is, 
perhaps,  much  cracked  up  because  of  the  extremes  of  heat 


204    AN   INTRODUCTION    TO   ASTRONOMY    [ch.  vn,  131 

and  cold  to  which  it  is  subject.  But  there  is  nothing  re- 
sembhng  soil  except,  possibly,  volcanic  ashes.  There  can  be 
no  water  on  the  moon ;  for,  if  there  were,  it  would  be  at  least 
partly  evaporated,  especially  in  the  long  day,  and  form  an 
atmosphere. 

One  cannot  refrain  from  asking  why  the  moon  has  no 
atmosphere.  It  may  be  that  it  never  had  any.  But  the 
evidence  of  great  surface  disturbances  makes  it  not  altogether 
improbable  that  vast  quantities  of  vapors  have  been  emitted 
from  its  interior.  If  this  is  true,  they  seem  to  have  dis- 
appeared. There  are  two  ways  in  which  their  disappearance 
can  be  explained.  One  is  that  they  have  xmited  chemically 
with  other  elements  on  the  moon.  As  a  possible  example  of 
such  action  it  may  be  mentioned  that  there  are  vast  quan- 
tities of  oxygen  in  the  rocks  of  the  earth's  crust,  which  may, 
perhaps,  have  been  largely  derived  from  the  atmosphere. 
The  second  explanation  is  that,  according  to  the  kinetic 
theory  of  gases,  the  moon  may  have  lost  its  atmosphere  by 
the  escape  of  molecule  after  molecule  from  its  gravitative 
control.  This  might  be  a  relatively  rapid  process  in  the  case 
of  a  body  having  the  low  velocity  of  escape  of  1.5  miles  per 
second  (Art.  33),  especially  if  its  days  were  so  long  that  its 
surface  became  highly  heated. 

It  seems  probable,  therefore,  that  the  moon  could  not 
retain  an  atmosphere  if  it  had  one,  and  that  whatever  gases 
it  may  ever  have  acquired  from  volcanoes  or  other  sources 
were  speedily  lost. 

132.  The  Light  and  Heat  received  by  the  Earth  from  the 
Moon.  —  The  average  distances  of  the  earth  and  the  moon 
from  the  sun  are  about  the  same ;  and,  consequently,  the 
earth  and  the  moon  receive  about  equal  amounts  of  light 
and  heat  per  unit  area.  The  amount  of  light  and  heat  that 
the  earth  receives  from  the  moon  depends  upon  how  much 
the  moon  receives  from  the  sun,  what  fraction  it  reflects, 
its  distance  from  the  earth,  and  its  phase.  It  is  easy  to  see 
that,  if  all  the  light  the  moon  receives  were  reflected,  the 


CH.  VII,  133]  THE   MOON  205 

amount  which  strikes  the  earth  could  be  computed  for  any 
phase  as,  for  example,  when  the  moon  is  full.  It  is  found  by- 
taking  into  account  all  the  factors  involved  that,  if  the  moon 
were  a  perfect  mirror,  it  would  give  the  earth,  when  it  is 
full,  about  loofooo  as  much  Ught  as  the  earth  receives  from 
the  sun.  As  a  matter  of  fact,  the  moon  is  by  no  means  a 
perfect  reflector,  and  the  amount  of  hght  it  sends  to  the 
earth  is  very  much  less  than  this  quantity. 

It  is  not  easy  to  compare  moonUght  with  sunhght  by  direct 
measurements,  and  the  results  obtained  by  different  observ- 
ers are  somewhat  divergent.  The  m'easurements  of  Zoll- 
ner,  which  are  commonly  accepted,  show  that  sunUght  is 
618,000  times  greater  than  the  Ught  received  from  the  full 
moon.  Sir  John  Herschel's  observations  gave  the  notably 
smaller  ratio  of  465,000.  At  other  phases  the  moon  gives 
not  only  correspondingly  less  Ught,  but  less  than  would  be 
expected  on  the  basis  of  the  part  of  the  moon  illuminated. 
For  example,  at  first  quarter  the .  illuminated  area  is  half 
that  at  full  moon,  but  the  amount  of  Ught  received  is  less 
than  one  eighth  that  at  full  moon.  This  phenomenon  is 
doubtless  due  to  the  roughness  of  the  moon's  surface.  More- 
over, the  amoimt  of  Ught  received  from  the  moon  near  first 
quarter  is  somewhat  greater  than  that  received  at  the  cor- 
responding phase  at  third  quarter,  the  difference  being  due 
to  the  dark  spots  on  the  eastern  limb  of  the  moon.  On 
taking  into  consideration  the  whole  month,  the  average 
amount  of  Ught  and  heat  which  the  moon  furnishes  the  earth 
cannot  exceed  2,500,000  o^  ^^^^  received  from  the  sun.  In 
other  terms,  the  earth  receives  as  much  Ught  and  heat  from 
the  sun  in  13  seconds  as  it  receives  from  the  moon  in  the 
course  of  a  whole  year. 

133.  The  Temperature  of  the  Moon.  —  The  temperature 
of  the  moon  depends  upon  the  amount  of  heat  it  receives, 
the  amount  it  reflects,  and  its  rate  of  radiation.  About  7 
per  cent  of  the  heat  which  falls  on  the  moon  is  directly  re- 
flected, and  this  has  no  effect  upon  its  temperature.     The 


206    AN   INTRODUCTION   TO  ASTRONOMY    [ch.  vii,  133 

remaining  93  per  cent  is  absorbed  and  raises  the  tempera- 
ture of  its  surface.  The  rate  of  radiation  of  the  moon's 
surface  materials  for  a  given  temperature  is  not  known  be- 
cause of  the  uncertainties  of  their  composition  and  physi- 
cal condition.  Nevertheless,  it  can  be  determined,  at  least 
roughly,  at  the  time  of  a  total  eclipse  of  the  moon. 

Consider  the  moon  when  it  is  nearly  full  and  just  before  it 
is  echpsed  by  passing  into  the  earth's  shadow,  as  at  N, 
Fig.  81.  The  side  toward  the  earth  is  subject  to  the  per- 
pendicular rays  of  the  sun  and  has  a  higher  temperature 
than  any  other  part  of  its  surface.  It  is  easy  to  measure 
with  some  approximation  the  amoxmt  of  heat  received  from 
the  moon,  but  it  is  not  easy  to  determine  what  part  of  it  is 
reflected  and  what  part  is  radiated.  Now  suppose  the  moon 
passes  on  into  the  earth's  shadow  so  that  the  direct  rays  of 
the  sun  are  cut  off.  Then  all  the  heat  received  from  the 
moon  is  that  radiated  from  a  surface  recently  exposed  to  the 
sun's  rays.  This  can  be  measured ;  and,  from  the  amoxmt 
received  and  the  rate  at  which  it  decreases  as  the  eclipse 
continues,  it  is  possible  to  determine  approximately  the 
rate  at  which  the  moon  loses  heat  by  radiation,  and  from 
this  the  temperature  to  which  it  has  been  raised.  The  obser- 
vations show  that  the  amount  of  heat  received  from  the 
moon  diminishes  very  rapidly  after  the  moon  passes  into 
the  earth's  shadow.  This  means  that  its  radiation  is  very 
rapid  and  that  probably  its  temperature  does  not  rise  very 
high.  It  doubtless  is  safe  to  state  that  at  its  maximima  it 
is  between  the  freezing  and  the  boiling  points.  The  recent 
work  of  Very  leads  to  the  conclusion  that  the  surface  is 
heated  at  its  highest  to  a  temperature  of  200°  Fahrenheit. 

It  is  now  possible  to  get  a  more  or  less  satisfactory  idea 
of  the  temperature  conditions  of  the  moon.  It  must  be 
remembered,  in  the  first  place,  that  its  day  is  28.5  times  as 
long  as  that  of  the  earth.  In  the  second  place,  it  has  no 
atmospheric  envelope  to  keep  out  the  heat  in  the  daytime 
and  to  retain  it  at  night.     Consequently,  when  the  sun  rises 


CH.  VII,  135]  THE   MOON  207 

for  a  point  on  the  moon,  its  rays  continue  to  beat  down 
upon  the  surface,  which  is  entirely  unprotected  by  clouds  or 
air,  for  more  than  14  of  our  days.  During  this  time  the 
temperature  rises  above  the  freezing  pbint  and  it  may  even 
go  up  to  the  boihng  point.  When  the  sun  sets,  the  darkness 
of  midnight  immediately  follows  because  there  is  no  atmos- 
phere to  produce  t^light,  and  the  heat  rapidly  escapes 
into  space.  In  the  course  of  an  hour  or  two  the  temperature 
of  the  surface  probably  falls  below  the  freezing  point,  and 
in  the  course  of  a  day  or  two  it  may  descend  to  100°  below 
zero.  It  will  either  remain  there  or  descend  still  lower  imtil 
the  sun  rises  again  14  days  after  it  has  set. 

The  cUmatic  conditions  on  the  moon  illustrate  in  the  most 
striking  manner  the  effects  of  the  earth's  atmosphere  and 
the  consequences  of  the  earth's  short  period  of  rotation. 

134.  General  surface  Conditions  on  the  Moon.  —  On  the 
whole,  the  surface  of  the  moon  is  extremely  rough,  showing 
no  effects  of  weathering  by  air  or  water.  It  is  broken  by 
several  moimtain  chains,  by  numerous  isolated  mountain 
peaks,  and  by  more  than  30,000  observed  craters.  There 
are  several  large,  comparatively  smooth  and  level  areas, 
which  were  called  maria  (seas)  by  Gahleo  and  other  early 
observers,  and  the  names  are  still  retained  though  modern 
instruments  show  that  they  not  only  contain  no  water  but 
are  often  rather  rough.'  The  smooth  places  are  the  areas 
which  are  relatively  dark  as  seen  with  the  xmaided  eye  or 
through  a  small  telescope.  For  example,  the  dark  patch 
near  the  bottom  of  Fig.  75  and  a  Httle  to  the  left  of  the 
center  with  a  rather  sharply  defined  lower  edge  is  known  as 
Mare  Serenitatis  (The  Serene  Sea).  The  fight  line  j-unning 
out  from  the  right  of  it  and  just  under  the  big  crater  Coper- 
nicus is  the  Apennine  range  of  mountains.  The  most  con- 
spicuous features  which  are  visible  with  an  ordinary  invert- 
ing telescope  are  shown  on  the  map.  Fig.  76. 

135.  The  Mountains  on  the  Moon. —  There  are  ten 
ranges  of  mountains  on  the  part  of  the  moon  which  is  visible 


208    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vu,  135 


from  the  earth.  The  mountains  are  often  extremely  slender 
and  lofty,  in  some  cases  attaining  an  altitude  of  more  than 
20,000  feet  above  the  plains  on  which  they  stand.     If  the 


Fig.  75.  —  The  moon  at  9|  days.     Photographed  at  the  Yerkes  Observatory. 


mountains  on  the  earth  were  relatively  as  large,  they  would 
be  more  than  15  miles  high.  The  height  of  the  lunar  moun- 
tains is  undoubtedly  due,  at  least  in  part,  to  the  low  surface 


CH.  VII,  135] 


THE   MOON 


209 


gravity  on  the  moon,  and  to  the  fact  that  there  has  been 
no  erosion  by  air  and  water. 

The  height  of  a  lunar  mountain  is  determined  from  the 
length  of  its  shadow  when  the  sun's  rays  strike  it  obliquely. 


BiW'rt^ 


^At6. S 


Fig.  76.  —  Outline  map  of  the  moon. 


For  example,  in  Fig.  77  the  crater  Theophilus  is  a  little 
below  the  center,  and  in  its  interior  are  three  lofty  moun- 
tains whose  sharp,  spirelike  shadows  stretch  off  to  the  left. 
Since  the  size  of  the  moon  and  the  scale  of  the  photograph 
are  both  known,  the  lengths  of  the  shadows  can  easily  be 


210    AN   INTRODUCTION   TO  ASTRONOMY    [ch.  vn,  135 


Fig.  77.  —  The  crater  Theophilus  and  surrounding  region  (Ritohey). 


determined.  There  is  also  no  difficulty  in  finding  the  height 
of  the  sun  in  the  sky  as  seen  from  this  position  on  the  moon 
when  the  picture  was  taken.  Consequently,  it  is  possible 
from  these  data  to  compute  the  height  of  the  mountains. 


CH.  VII,  136]  THE   MOON  211 

In  the  particular  case  of  Theophilus,  the  mountains  in  its 
interior  are  more  than  16,000  feet  above  its  floor.  On  the 
earth  the  heights  of  mountains  are  counted  from  the  sea 
level,  which,  in  most  cases,  is  far  away.  For  example.  Pike's 
Peak  is  about  14,000  feet  above  the  level  of  the  ocean,  which 
is  more  than  1000  miles  away,  but  only  about  half  that 
height  above  the  plateau  on  which  it  rests.  The;  shadows 
of  the  lunar  moimtains  are  black  and  sharp  because  the 
moon  has  no  atmosphere,  and  they  are  therefore  well  suited 
fpr  use  in  measuring  the  heights  of  objects  on  its  surface.. 

136.  Lunar  Craters.  —  The  most  remarkable  and  the  most 
conspicuous  objects  of  the  lunar  topography  are  the  craters,, 
of  which  more  than  30,000  have  been  mapped.  There  have 
been  successive  stages  in  their  formation,  for  new  ones  in 
many  places,  have  broken  through  and  encroached  upon  the 
old,  as  shown  in  Pig.  78.  Sometimes  the  newer  ones  are 
precisely  on  the  rims  of  the  older,  and  sometimes  they  are 
entirely  in  their  interiors.  The  newer  craters  have  deeper 
floors  and  steeper  and  higher  rims  than  the  older,  and  one 
of  the  most  interesting  things  about  them  is  that  very  often 
they  have  near  their  centers  lofty  and  spireUke  peaks. 

The  term  crater  at  once  carries  the  impression  to  the  mind 
that  these  objects  on  the  moon  are  analogous  to  the  vol- 
canic craters  on  the  earth.  There  is  at  least  an  immense 
difference  in  their  dimensions.  Many  lunar  craters  are  from 
50  to  60  miles  in  diameter,  and,  in  a  number  of  cases,  their 
diameters  exceed  100  miles.  Ptolemy  is  115  miles  across, 
while  Theophilus  is  64  miles  in  diameter  and  19,000  feet  deep. 
The  lofty  peak  in  the  great  crater  Copernicus  towers  11,000 
feet  above  the  plains  from  which  it  rises.  Some  of  these 
craters  are  on  such  an  enormous  scale  that  their  rims  would 
not  be  visible  from  their  centers  because  of  the  curvature  of 
the  surface  of  the  moon. 

The  explanation  of  the  craters  is  by  no  means  easy,  and 
universal  agreement  has  not  been  reached.    If  they  are  of 


212    AN   INTRODUCTION   TO  ASTRONOMY    [ch.  vii,  136 


f./,^  «>'■-■"  ';y 


Fig.  78.  —  The  great  crater  Clavius  with  smaller  craters  on  its  rim 
and  in  its  interior.  Photographed  by  Ritchey  with  the  4.0-inch  telescope 
of  the  Yerkes  Observatory. 


CH.  VII,  136]  THE   MOON  213 

volcanic  origin,  the  activity  which  was  present  on  the  moon 
enormously  surpassed  anything  now  known  on  the  earth. 
In  view  of  the  fact  that  there  are  no  lava  flows,  and  that  in 
most  cases  the  material  around  a  crater  would  not  fill  it, 
the  volcanic  theory  of  their  origin  seems  very  improbable 
and  has  been  abandoned.  Another  suggestion  is  that  the 
craters  have  been  formed  by  the  bursting  out  of  great  masses 
of  gas  which  gathered  under  the  surface  of  the  moon  and 
became  heated  and  subject  to  great  tension  because  of  its 
contraction.  According  to  this  theory,  the  escaping  gas 
threw  out  large  masses  of  the  material  which  covered  it  and 
thus  made  the  rims  of  the  craters.  But  it  is  hard  to  account 
for  the  mountains  which  are  so  often  seen  in  the  interiors 
of  craters. 

Gilbert  suggested  that  the  limar  craters  may  have  been 
formed  by  the  impacts  of  huge  meteorites,  in  some  cases  many 
miles  across.  It  is  certain  that  such  bodies,  weighing  hun- 
dreds of  pounds  and  even  tons,  now  fall  upon  the  earth 
occasionally.  It  is  supposed  that  milhons  of  years  ago  the 
collisions  of  these  wandering  masses  with  the  earth  and 
moon  were  much  more  frequent  than  they  are  at  the 
present  time.  When  they  strike  the  earth,  their  energy  is 
largely  taken  up  by  the  cushion  of  the  earth's  atmosphere ; 
when  they  strike  the  moon,  they  plimge  in  upon  its  surface 
with  a  speed  from  50  to  100  times  that  of  a  cannon  ball. 
It  does  not  seem  improbable  that  masses  many  miles  across 
and  weighing  milhons  of  tons  might  produce  splashes  in  the 
surface  of  the  moon,  even  though  it  be  sohd  rock,  analo- 
gous to  the  craters  which  are  now  observed.  The  heat 
generated  by  the  impacts  would  be  sufficient  to  liquefy  the 
materials  immediately  under  the  place  where  the  meteorites 
struck,  and  might  even  cause  very  great  explosions.  The 
mountains  in  the  centers  might  be  due  to  a  sort  of  re- 
action from  the  original  splash,  or  from  the  heat  produced 
by  the  colhsion.  At  any  rate,  numerous  experiments  with 
projectiles  on  a  variety  of  substances  have  shown  that  pits 


214    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  136 

closely  resembling  the  lunar  craters  are  very  often  obtained. 
This  view  as  to  the  cause  of  the  craters  is  in  harmony  with 
the  theory  that  the  earth  and  moon  grew  up  by  the  accre- 
tion of  widely  scattered  material  around  nuclei  which  were 
originally  of  much  smaller  dimensions  (Art.  250). 

An  obvious  objection  to  the  theory  that  the  craters  on 
the  moon  were  produced  by  meteorites  is  that  the  earth  has 
no  similar  formations.  Since  the  earth  and  moon  are  closely 
associated  in  their  revolution  around  the  sim,  it  is  clear  that 
the  earth  would  have  been  bombarded  at  least  as  violently 
as  the  moon.  The  answer  to  this  objection  is  that,  for  mil- 
lions of  years,  the  rains  and  snows  and  atmosphere  have  dis- 
integrated the  craters  and  mountains  on  the  earth,  and  their 
powdered  remains  have  been  carried  away  into  the  valleys. 
Whatever  irregularities  of  this  character  the  earth's  surface 
may  have  had  in  its  early  stages,  all  traces  of  them  disap- 
peared millions  of  years  ago.  On  the  other  hand,  since  air 
and  water  are  altogether  absent  from  the  moon,  this  nearest 
celestial  body  has  preserved  for  us  the  records  of  the  forces 
to  which  it,  and  probably  also  the  earth,  were  subject  in  the 
early  stages  of  their  development. 

Probably  the  most  serious  objection  to  the  impact  theory 
of  the  craters  on  the  moon  is  that  they  nearly  all  appear  to 
have  been  made  by  bodies  falling  straight  toward  the  moon's 
center.  It  is  obvious  that  a  sphere  circulating  in  space 
would  in  a  majority  of  cases  be  struck  glancing  blows  by 
wandering  meteorites.  The  attraction  of  the  moon  would 
of  course  tend  to  draw  them  toward  its  center,  but  their 
velocities  are  so  great  that  this  factor  cannot  seriously 
have  modified  their  motions.  The  only  escape  from  this 
objection,  so  far  as  suggested,  is  that  the  heat  generated  by 
the  impacts  may  have  been  sufficient  to  hquefy  the  material 
in  the  neighborhood  of  the  places  where  the  meteorites  struck, 
and  thus  to  destroy  all  evidences  of  the  directions  of  the  blows. 

137.  Rays  and  Rills.  —  Some  of  the  large  craters,  par- 
ticularly Tycho  and  Copernicus,  have  long  hght  streaks, 


CH.  VII,  137]  THE   MOON  215 

called  rays,  radiating  from  them  like  spokes  from  the  axle  of 
a  wheel.  They  are  not  interfered  with  by  hill  or  valley, 
and  they  often  extend  a  distance  of  several  hundred  miles. 
They  cast  no  shadows,  which  proves  that  they  are  at  the 
same  level  as  the  adjacent  surface,  and  they  are  most  con- 


■pp 

1 

^1 

■^', 

^ 

■if 

^^ 

n 

^Kt  '^"'^  i^^Ih^^I 

V'j 

^^^^HHBt     ^TiS^ 

Bk  ^i 

^^^n^^'  i^H-^' 

'^^^yHHH^^H^^^^l 

Hr  .^^1 

^^^^^■^  ^^HH^^ 

s 

m 

Fig.  79.  —  The  full  moon.     Photographed  at  the  Yerkes  Observatory  (Wallace). 

spicuous  at  the  time  of  full  moon.  They  are  easily  seen  in 
Pig.  79.  It  has  been  supposed  by  some  that  they  are  lava 
streams  and  by  others  that  they  were  great  cracks  in  the 
surface,  formed  at  the  time  when  the  craters  were  produced, 
which  have  since  filled  up  with  Ughter  colored  material 
from  below. 


216    AN   ESTTRODUCTION   TO   ASTRONOMY    [ch.  vii,  137 

The  rills  are  cracks  in  the  moon's  surface,  a  mile  or  so 
wide,  a  quarter  of  a  mile  deep,  and  sometimes  as  much  as 
150  miles  in  length.  They  are  very  numerous,  more  than  • 
1000  having  been  so  far  mapped.  The  only  things  at  all 
hke  them  on  the  earth  are  such  chasms  as  the  Grand  Can- 
yon of  the  Colorado  and  the  cut  below  Niagara  Falls.  But 
these  gorges  are  the  work  of  erosion,  which  has  probably 
beeif  entirely  absent  from  the  surface  of  the  moon.  At  any 
rate,  it  is  incredible  that  the  rills  have  been  produced  by 
erosion.  The  most  plausible  theory  is  that  they  are  cracks 
which  have  been  caused  by  violent  volcanic  action,  or  by 
the  rapid  cooling  and  shrinking  of  the  moon. 

The  rays  and  rills  are  very  puzzling  lunar  features  which 
seem  to  be  fundamentally  unlike  anything  in  terrestrial 
topography.  Even  our  nearest  neighbor  thus  differs  very 
radically  from  the  earth. 

138.  The  Question  of  Changes  on  the  Moon.  —  There 
have  been  no  observed  cl^anges  in  the  larger  features  of  the 
lunar  topography,  although,  from  time  to  time,  minor  alter- 
ations have  been  suspected.  The  most  probable  change  of 
any  natural  physical  feature  is  in  the  small  crater  Linn6,  in 
Mare  Serenitatis.  It  was  mapped  about  a  century  ago, 
but  in  1866  was  said  by  Schmidt  to  be  entirely  invisible. 
It  is  now  visible  as  on  the  original  maps.  It  is  generally 
beUeved  that  the  differences  in  appearance  at  various  times 
have  been  due  to  slightly  different  conditions  of  illumination. 

Since  the  moon's  orbit  is  constantly  shifting  because  of 
the  attraction  of  the  sun,  and  since  the  month  does  not  con- 
tain an  integral  number  of  days,  it  follows  that  an  observer 
never  gets  at  two  different  times  exactly  the  same  view  of 
the  moon.  W.  H.  Pickering  has  noticed  changes  in  some 
small  craters,  depending  upon  the  phase  of  the  moon,  which 
he  interprets  as  possibly  being  due  to  some  kind  of  vegeta- 
tion which  flourishes  in  the  valleys  where  he  supposes  heavier 
gases,  such  as  carbon  dioxide,  might  collect.  Some  of  his 
observations  have  been  verified  by  other  astronomers,  but 


CH.  VII,  139]  THE   MOON  217 

his  rather  bold  speculations  as  to  their  meaning  have  not 
been  accepted. 

It  is  altogether  probable  that  the  moon  long  ago  arrived 
at  the  stage  where  surface  changes  practically  ceased.  The 
only  known  influences  which  could  now  disturb  its  surface 
are  the  feeble  tidal  strains  to  which  it  is  subject,  and  the 
extremes  of  temperature  between  night  and  day.  While  it 
would  be  too  much  to  say  that  slight  disintegration  of  the 
surface  rocks  may  not  still  be  taking  place,  yet  it  is  certain 
that,  on  the  whole,  the  moon  is  a  body  whose  evolution  is 
essentially  finished.  The  seasonal  changes  are  unimportant, 
but  there  is  alternately  for  two  weeks  the  bUnding  glare  of 
the  sunlight,  never  tempered  by  passing  clouds  or  even  an 
atmosphere,  and  the  blackness  and  frigidity  of  the  long  lunar 
night.  Month  succeeds  month,  age  after  age,  with  no  im- 
portant variations  in  these  phenomena. 

139.  The  Effects  of  the  Moon  on  the  Earth.  —  The  moon 
reflects  a  relatively  small  amount  of  sunUght  and  heat  to 
the  earth,  and  in  conjunction  with  the  sun  it  produces  the 
tides.  These  are  the  only  influences  of  the  moon  on  the 
earth  that  can  be  observed  by  the  ordinary  person.  It  has 
a  number  of  very  minor  effects,  such  as  causing  minute 
variations  in  the  magnetic  needle,  the  precession  of  the  equi- 
noxes, and  slight  changes  in  the  motion  of  the  earth;  but 
they  are  all  so  small  that  they  can  be  detected  only  by  re- 
fined scientific  methods. 

There  are  a  great  many  ideas  popularly  entertained,  such 
as  that  it  is  more  liable  to  rain  at  the  time  of  a  change  of 
the  moon,  or  that  crops  grow  best  when  planted  in  certain 
phases,  which  have  no  scientific  foundation  whatever.  It 
follows  from  the  fact  that  more  light  and  heat  are  received 
from  the  sun  in  13  seconds  than  from  the  moon  in  a  whole 
year,  that  its  heating  effects  on  the  earth  cannot  be  impor- 
tg-nt.  The  passing  of  a  fleecy  cloud,  or  the  haze  of  Indian 
summer,  cuts  off  more  heat  from  the  sun  than  the  moon 
sends  to  the  earth  in  a  year.     Consequently,  it  is  entirely 


218    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  139 

unreasonable  to  suppose  that  the  moon  has  any  important 
climatic  effects  on  the  earth.  Besides  this,  recorded  obser- 
vations of  temperature,  the  amount  of  rain,  and  the  velocity 
of  the  wind,  in  many  places,  for  more  than  100  years,  fail 
to  show  with  certainty  any  relation  between  the  weather  and 
phases  of  the  moon. 

The  phenomena  of  storms  themselves  show  the  essential 
independence  of  the  weather  and  the  phases  of  the  moon. 
Storm  centers  move  across  the  country  in  a  northeasterly 
direction  at  the  rate  of  400  to  500  miles  per  day,  and  some- 
times they  can  be  followed  entirely  around  the  earth.  Con- 
sequently, if  a  storm  should  pass  one  place  at  a  certain  phase 
of  the  moon,  it  would  pass  another  a  few  thousand  miles 
eastward  at  quite  a  different  phase.  The  theory  that  a 
storm  occurred  at  a  certain  phase  of  the  moon  would  then 
be  verified  for  one  longitude  and  would  fail  of  verification 
at  all  the  others. 

140.  Eclipses  of  the  Moon.  —  The  moon  is  eclipsed  when- 
ever it  passes  into  the  earth's  shadow  so  that  it  does  not 


Fig.   80.  —  The  condition  for  eclipses  of  the  moon  and  sun. 

receive  the  direct  light  of  the  sun.  In  Fig.  80,  E  represents 
the  earth  and  PQR  the  earth's  shadow,  which  comes  to  a 
point  at  a  distance  of  870,000  miles  from  the  earth's  center. 
The  only  light  received  from  the  sun  within  this  cone  is 
that  small  amount  which  is  refracted  into  it  by  the  earth's 
atmosphere  in  the  zone  QR.  In  the  regions  TQP  and  SRP 
the  sun  is  partially  ecUpsed,  the  light  being  cut  off  more  and 
more"  as  the  shadow  cone  is  approached.  The  shadow  cone 
PQR  is  called  the  umbra,  and  the  parts  TQP  and-<8jBP,  the 
penumbra. 


CH.  vii,  140]  THE   MOON  219 

When  the  moon  is  about  to  be  eclipsed,  it  passes  from  full 
illumination  by  the  sun  gradually  into  the  penumbra,  where 
at  first  only  a  small  part  of  the  sun  is  obscured,  and  it  then 
proceeds  steadily  across  the  shadow  of  increasing  density 
until  it  arrives  at  A,  where  the  sun's  Ught  is  entirely  cut  off. 
The  distance  across  the  earth's  shadow  is  so  great  that  the 
moon  is  totally  echpsed  for  nearly  2  hours  while  it  is  pass- 
ing through  the  umbra,  and  the  time  from  the  first  contact 
with  the  umbra  until  the  last  is  about  3  hours  and  45  minutes. 

It  appears  from  Fig.  80  that  the  moon  would  be  echpsed 
every  time  it  is  in  opposition  to  the  sun,  but  this  figure  is 
drawn  to  show  the  relations  as  one  looks  perpendicularly 
on  the  plane  of  the  echptic,  neglecting  the  inclination  of  the 
moon's  orbit.     Figure  81  shows  another  section  in  which 


Fig.  81.  —  Condition  in  which  eclipses  of  the  moon  and  sun  fail. 

the  plane  of  the  moon's  orbit,  represented  by  MiV,  is  per- 
pendicular to  the  page.  It  is  obvious  from  this  that,  when 
the  moon  is  in  the  neighborhood  of  iV,  it  will  pass  south  of 
the  earth's  shadow  instead  of  through  it.  The  proportions 
in  the  figure  are  by  no  means  true  to  scale,  but  a  detailed 
discussion  of  the  numbers  involved  shows  that  usually  the 
moon  will  pass  through  opposition  to  the  sun  without  en- 
countering the  earth's  shadow.  But  when  the  earth  is  90° 
in  its  orbit  from  the  position  shown  in  the  figure,  that  is, 
when  the  earth  as  seen  from  the  sun  is  at  a  node  of  the  moon's 
orbit,  the  plane  of  the  moon's  orbit  will  pass  through  the 
sun,  and  consequently  'the  moon  will  be  eclipsed.  At  least, 
the  moon  will  be  echpsed  if  it  is  full  when  the  earth  is  at  or 
near  the  node.  The  earth  is  at  a  node  of  the  moon's  orbit 
at  two  times  in  the  year  separated  by  an  interval  of  six 


220    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  140 

months.  Consequently,  there  may  be  two  eclipses  of  the 
moon  a  year ;  but  because  the  moon  may  not  be  full  when 
the  earth  is  at  one  of  these  positions,  one  or  both  of  the 
eclipses  may  be  missed. 

Since  the  sun  apparently  travels  along  the  ecliptic  in  the 
sky,  the  earth's  shadow  is  on  the  ecUptic  180°  from  the  sun. 
The  places  where  the  moon  crosses  the  ecliptic  are  the 
nodes  of  its  orbit,  and,  consequently,  there  can  be  an  echpse 
of  the  moon  only  when  it  is  near  one  of  its  nodes.  Since 
the  nodes  continually  regress  as  a  consequence  of  the  sun's 
attraction  for  the  moon,  the  eclipses  occur  earlier  year  after 
year,  completing  a  cycle  in  18.6  years. 

One  scientific  use  of  eclipses  of  the  moon  is  that  when  they 
occur,  the  heat  radiated  by  the  moon  after  it  has  just  been 
exposed  to  the  perpendicular  rays  of  the  sun  gives  an  op- 
portunity, as  was  explained  in  Art.  133,  of  determining  its 
temperature.  Also,  at  the  time  of  a  lunar  eclipse,  the  stars 
in  the  neighborhood  of  the  moon  can  easily  be  observed,  and 
it  is  a  simple  matter  to  determine  the  exact  instant  at  which 
the  moon  passes  in  front  of  a  star  and  cuts  off  its  Ught. 
Since  the  positions  of  the  stars  are  well  known,  such  an 
observation  locates  the  moon  with  great  exactness  at  the 
time  the  observation  is  made.  It  is  imaginable  that  the 
moon  may  be  attended  by  a  small  satelUte.  If  the  moon  is 
not  echpsed,  its  own  light  or  that  of  the  sim  will  make  it 
impossible  to  see  a  very  minute  body  in  its  neighborhood; 
but  at  the  time  of  an  echpse,  a  satelhte  may  be  exposed  to 
the  rays  of  the  sun  while  the  neighboring  sky  will  not  be 
hghted  up  by  the  moon.  Only  at  such  a  time  would  there 
be  any  hope  of  discovering  a  small  body  revolving  around 
the  moon.  A  search  for  such  an  attendant  has  been  made, 
but  has  so  far  proved  fruitless. 

141.  Eclipses  of  the  Sun.  —  The  sun  is  echpsed  when 
the  moon  is  so  situated  as  to  cut  off  the  sun's  Ught  from  at 
least  a  portion  of  the  earth.  The  apparent  diameter  of  the 
moon  is  only  a  httle  greater  than  that  of  the  sun,  and,  con- 


CH.  VII,  141] 


THE   MOON 


221 


sequently,  eclipses  of  the  sun  last  for  a  very  short  time. 
This  statement  is  equivalent  to  saying  that  the  shadow  cone 
of  the  moon  comes  to  a  point  near  the  surface  of  the  earth, 
as  is  shown  in  Fig.  80.  It  is  also  obvious  from  this  diagram 
that  the  sun  is  eclipsed  as  seen  from  only  a  small  part  of  the 
earth.  As  the  moon  moves  around  the  earth  in  its  orbit 
and  the  earth  rotates  on  its  axis,  the  shadow  cone  of  the 
moon  describes  a  streak  across  the  earth  which  may  be 
somewhat  curved. 

It  follows  from  the  fact  that  the  path  of  the  moon's  shadow 
across  the  earth  is  very  narrow,  as  shown  in  Fig.  82,  that  a 


Fig.  82.  —  Path  of  the  total  eclipse  of  the  sun,  August  29-30,  1905. 

total  ecUpse  of  the  sun  will  be  observed  very  infrequently 
at  any  given  place.  On  this  account,  as  well  as  because  it 
is  a  startUng  phenomenon  for  the  sun  to  become  dark  in  the 
daytime,  eclipses  have  always  been  very  noteworthy  occur- 
rences. Repeatedly  in  ancient  times,  in  which  the  chro- 
nology was  very  uncertain,  writers  referred  toecUpses  in  con- 
nection with  certain  historical  events,  and  astronomers, 
calculating  back  across  the  centuries,  have  been  able  to 


222    AN   INTRODUCTION   TO   ASTRONOMY    [ch.  vii,  141 

identify  the  eclipses  and  thus  fix  the  dates  for  historians  in 
the  present  system  of  counting  time.  The  infrequency  of 
eclipses  at  any  particular .  place  is  evident  from  Fig.  83, 
which  gives  the  paths  of  all  the  total  eclipses  of  the  sun 
from  1894-1973.  In  this  long  period  the  greater  part  of 
the  world  is  not  touched  by  them  at  all. 

So  far  the  discussion  has  referred  only  to  total  ecUpses  of 
the  sun ;  but  in  the  regions  on  the  earth's  surface  which  are 


BORUAy  It  CO^  h.r.. 

Fig.  83.  —  Paths  of  total  eclipses  of  the  sun.      (From  Todd's  Total  Eclipses.) 


near  the  path  of  totality,  or  in  the  penumbra  -of  the  moon's 
shadow,  which  is  entirely  analogous  to  that  of  the  earth, 
there  are  partial  eclipses  of  the  sun.  The  region  covered  by 
the  penumbra  is  many  times  that  where  an  eclipse  is  total ; 
and,  consequently,  partial  eclipses  of  the  sun  are  not  very 
infrequent  phenomena. 

There  is  not  an  eclipse  of  the  sun  every  time  the  moon  is 
in  conjimction  with  the  sun  because  of  the  inclination  of  its 
orbit.     For  example,  when  it  is  near  M,  Fig.  81,  its  shadow 


CH.  VII,  142]  THE   MOON  223 

passes  north  of  the  earth.  In  fact,  ecUpses  of  the  sun  occur 
only  when" the  sun  is  near  one  of  the  moon's  nodes,  just  as 
ecUpses  of  the  moon  occur  only  when  the  earth's  shadow  is 
near  one  of  the  moon's  nodes.  Consequently,  eclipses  occur 
twice  a  year  at  intervals  separated  by  6  synodical  months. 
Since  the  moon's  nodes  regress,  making  a  revolution  in  18.6 
years,  echpses  occur,  on  the  average,  about  20  days  earher 
each  year  than  on  the  preceding  year. 

The  distance  UV,  Fig.  80,  within  which  an  ecUpse  of  the 
sun  can  occur  is  greater  than  AB,  within  which  an  echpse 
of  the  moon  can  occur.  Therefore  it  is  not  necessary  that 
the  sun  shall  be  as  near  the  moon's  node  in  order  that  an 
echpse  of  the  sun  may  result  as  it  is  in  order  that  there  may 
be  an  echpse  of  the  moon.  When  the  relations  are  worked 
out  fully,  it  is  found  that  there  will  be  at  least  one  solar 
eclipse  each  time  the  sun  passes  the  moon's  node,  and  that 
there  may  be  two  of  them.  Consequently,  in  a  year,  there 
may  be  two,  three,  or  four  eclipses  of  the  sun.  If  there  are 
only  two  echpses,  the  moon's  shadow  is  hkely  to  strike 
somewhere  near  the  center  of  the  earth  and  give  a  total 
echpse.  On  the  other  hand,  if  there  are  two  eclipses  while 
the  sun  is  passing  a  single  node  of  the  moon's  orbit,  they 
must  occur,  one  when  the  sun  is  some  distance  from  the  node 
on  one  side,  and  the  other  when  it  is  some  distance  from  the 
node  on  the  other  side.  In  this  case  the  moon's  shadow, 
or.  at  least  its  penumbra,  strikes  first  near  one  pole  of  the 
earth  and  then  near  the  other.  These  eclipses  are  generally 
only  partial.    ' 

142.  Phenomena  of  total  Solar  Eclipses.  —A  total  echpse 
of  the  sun  is  a  startling  phenomenon.  It  alv/ays  occurs  pre- 
cisely at  new  moon,  and  consequently  the  moon  is  invisible 
until  it  begins  to  obscure  the  sun.  The  first  indication  of  a 
solar  eclipse  is  a  black  sht  or  section  cut  out  of  the  western 
edge  of  the  sun  by  the  moon  which  is  passing  in  front  of  it 
from  west  to  east.  For  some  time  the  sunhght  is  not 
diminished  enough  to  be  noticeable.     Steadily  the  moon 


224    AN   INTRODUCTION   TO   ASTRONOMY     ch.  vii,  142 

moves  over  the  sun's  disk;  and,  as  the  instant  of  totality 
draws  near,  the  Hght  rapidly  fails,  animals  become  restless, 
and  everything  takes  on  a  weird  appearance.  Suddenly  a 
shadow  rushes  across  the  surface  of  the  earth  at  the  rate  of 
more  than  1300  miles  an  hour,  the  sun  is  covered,  the  stars 
flash  out,  around  the  apparent  edge  of  the  moon  are  rose- 
colored  prominences  (Art.  236)  of  vaporous  material  forced 
up  from  the  sun's  surface  to  a  height  of  perhaps  200,000 
miles,  and  all  around  the  sun,  extending  out  as  far  as  half 
its  diameter,  are  the  streamers  of  pearly  hght  which  con- 
stitute the  sun's  corona  (Art.  238).  After  about  7  minutes, 
at  the  very  most,  the  western  edge  of  the  sun  is  uncovered, 
dayhght  suddenly  reappears,  and  the  phenomena  of  a  partial 
ecUpse  take  place  in  the  reverse  order. 

Total  echpses  of  the  sun  afford  the  most  favorable  condi- 
tions for  searching  for  small  planets  within  the  orbit  of  Mer- 
cury, and  it  is  only  during  them  that  the  sun's  corona  can  be 
observed. 

X.    QUESTIONS 

1.  Verify  by  observations  the  motion  of  the  moon  eastward 
among  the  stars,  and  its  change  in  declination  during  a  month. 

2.  For  an  observer  on  the  moon  describe,  (a)  the  apparent 
motions  of  the  stars  ;  (6)  the  motion  of  the  sun  with  respect  to  the 
stars  ;  (c)  the  diurnal  motion  of  the  sun ;  (d)  the  motion  of  the  earth 
with  respect  to  the  stars ;  (e)  the  motion  of  the  earth  with  respect  to 
the  sun ;  (/)  the  diurnal  motion  of  the  earth  ;  [g)  the  librations  of  the 
earth. 

3.  Describe  the  phases  the  moon  would  have  throughout  the 
year  if  the  plane  of  its  orbit  were  perpendicular  to  the  plane  of  the 
eoliptic. 

4.  What  would  be  the  moon's  synodical  period  if  it  revolved 
around  the  earth  from  east  to  west  in  the  same  sidereal  period  ? 

5.  Show  by  a  diagram  that,  if  the  moon  always  presents  the  same 
face  toward  the  earth,  it  rotates  on  its  axis  and  its  period  of  rotation 
equals  the  sidereal  month. 

6.  Is  it  possible  that  the  moon  has  an  atmosphere  and  water  on 
the  side  remote  from  the  earth  ? 

7.  Suppose  you  could  go  to  the  moon  and  live  there  a  month. 


CH.  VII,  142]  THE   MOON  225 

Give  details  regarding  what  you  would  observe  and  the  experiences 
you  would  have. 

8.  What  are  the  objections  to  the  theory  that  lunar  craters  are 
of  volcanic  origin  ?    Tliat  they  were  produced  by  meteorites  ? 

9.  How  do  you  interpret  rays  and  rills  under  the  hypbthesis  that 
lunar  craters  were  produced  by  meteorites  ? 

10.  If  the  earth's  reflecting  power  is  4  times  that  of  the  moon, 
how  does  earthshine  on  the  moon  compare  with  moonshine  on  the 
earth  ? 


CHAPTER  VIII 
THE    SOLAR   SYSTEM 

I.   The  Law  of  Gravitation 

143.  The  Members  of  the  Solar  System.  —  The  members 
of  the  solar  system  are  the  sun,  the  planets  and  their  satel- 
lites, the  planetoids,  the  comets,  and  the  meteors.  It  may 
possibly  be  that  some  of  the  comets  and  meteors,  coming  in 
toward  the  sun  from  great  distances  and  passing  on  again, 
are  only  temporary  members  of  the  system.  The  sun  is 
the  one  preeminent  body.  Its  volume  is  nearly  a  thousand 
times  that  of  all  the  other  bodies  combined,  its  mass  is  so 
great  that  it  controls  all  their  motions,  and  its  rays  illuminate 
and  warm  them.  It  is  impossible  to  treat  of  the  planets 
without  taking  into  account  their  relations  to  the  sun,  but 
the  constitution  and  evolution  of  the  sun  are  quite  inde- 
pendent of  the  planets. 

The  eight  known  planets  are,  in  the  order  of  their  distance, 
from  the  sun,  Mercury,  Venus,  Earth,  Mars,  Jupiter,  Saturn, 
Uranus,  and  Neptune.  The  first  six  are  conspicuous  objects 
to  the  unaided  eye  when  they  are  favorably  located,  and  they 
have  been  known  from  prehistoric  times;  Uranus  and 
Neptune  were  discovered  in  1781  and  1846,  respectively. 
The  planetoids  (often  called  the  small  planets  and  sometimes 
the  asteroids)  are  small  planets  which,  with  a  few  exceptions, 
revolve  around  the  sun  between  the  orbits  of  Mars  and 
Jupiter.  The  comets  are  bizarre  objects  whose  orbits  are 
very  elongated  and  lie  in  every  position  with  respect  to  the 
orbits  of  the  planets.  Probably  at  least  a  part  of  the  meteors 
are  the  remains  of  disintegrated  comets;  they  are  visible 
■only  when  they  strike  into  the  earth's  atmosphere. 

226 


CH.  VIII,  144] 


THE   SOLAR  SYSTEM 


227 


144.  The  Relative  Dimensions  of  the  Planetary  Orbits.  — 

The  distance  from  the  earth  to  the  sun  is  called  the  astro- 
nomical unit.  The  distances  from  the  planets  to  the  sun  can 
be  determined  in  terms  of  the  astronomical  unit  without 
knowing  its  value  in  miles. 

Consider  first  the  planets  whose  orbits  are  interior  to  that 
of  the  earth.  They  are  called  the  inferior  planets.  In 
Fig.  84  let  S  represent  the  sun,  V  the  planet  Venus,  and 
E  the  earth.  The 
angle  SEV  is  called 
the  elongation  of  the 
planet,  and  may  vary 
from  zero  up  to  a 
maximiun  which  de- 
pends upon  the  size  of 
the  orbit  of  V.  When 
the  elongation  is  great- 
est, the  angle  at  y  is  a 
right  angle.  Suppose 
the  elongation  of  V  is 
determined  by  obser- 
vation day  after  day 
until  it  reaches  its 
maximum.  Then,  since 
the  elongation  is  measured  and  the  angle  at  V  is  90°,  the 
shape  of  the  triangle  is  determined,  and  SV  can  be  com- 
puted by  trigonometry  in  terms  of  SE, 

Now  consider  the  planets  whose  orbits  are  outside  that 
of  the  earth.  They  are  called  the  superior  planets.  Sup- 
pose the  periods  of  revolution  of  the  earth  and  Mars,  for 
example,  have  been  determined  from  long  series  of  obser- 
vations. This  can  be  done  without  knowing  anything  about 
their  actual  or  relative  distances.  For,  in  the  first  place, 
the  earth's  period  can  be  obtained  from  observations  of  the 
apparent  position  of  the  sun  with  respect  to  the  stars ;  and 
then  the  period  of  Mars  can  be  found  from  the  time  re- 


FiG.  84. 


-  Finding  the  distance  of  an 
inferior  planet. 


228    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  viii,  144 


quired  for  it  to  move  from  a  certain  position  with  respect 
to  the  sun  back  to  the  same  position  again.  For  example, 
when  a  planet  is  exactly  180°  from  the  sun  in  the  sky,  as 
seen  from  the  earth,  it  is  said  to  be  in  opposition.  The  period 
from  opposition  to  opposition  is  called  the  synodical  period 
(compare  Art.  120).  Let  the  sidereal  period  of  the  earth 
be  represented  by  E,  the  sidereal  period  of  the  planet  by  P, 
and  its  synodical  period  by  )S.  Then,  analogous  to  the  case 
of  the  moon  in  Art.  120,  P  is  defined  by 


1 
P 


1 

E 


s 


Now  return  to  the  problem  of  finding  the  distance  of  a 
superior   planet   in   terms   of   the   astronomical   unit.     In 

Fig.  85,  let  S  represent  the 
sun,  and  Ei  and  Mi  the 
positions  of  the  earth  and 
Mars  when  Mars  is  in  oppo- 
sition. Let  E2  and  M2  rep- 
resent the  positions  of  the 
earth  and  Mars  when  the 
angle  at  E2  is,  for  example, 
a  right  angle.  Mars  is  then 
said  to  be  in  quadrature,  and 
the  time  when  it  has  this 
position  can  be  determined 
by  observation.  The  angles 
M1SE2  and  M1SM2  can  be 
determined  from  the  periods  of  the  earth  and  Mars  and  the 
interval  of  time  required  for  the  earth  and  Mars  to  move 
from  El  and  Mi  respectively  to  E2  and  M2.  The  difference 
of  these  two  angles  is  M2SE2,  from  which,  together  with 
the  right  angle  at  E2,  the  distance  SM2  in  terms  of  SE2  can 
be  computed  by  trigonometry. 

A  httle  complication  in  the  processes  which  have  been 
described  arises  from  the  fact  that  the  orbit  of  the  earth  is 


Fig.  85.- 


-  Finding  the  distance  of  a 
superior  planet. 


CH.  VIII,  145] 


THE   SOLAR   SYSTEM 


229 


not  a  circle.  But  the  manner  in  which  the  distance  of  the 
earth  from  the  sun  varies  can  easily  be  determined  from 
observations  of  the  apparent  diameter  of  the  sun,  for  the 
apparent  diameter  of  an  object  varies  inversely  as  its  dis- 
tance. After  the  variations  in  the  earth's  distance  have  been 
found,  the  results  can  all  be  reduced  without  difficulty  to  a 
single  unit.  The  unit  adopted  is  half  the  length  of  the 
earth's  orbit,  and  is  called  its  mean  'distance,  though  it  is  a 
little  less  than  the  average  distance  to  the  sun. 

145.  Kepler's  Laws  of  Planetary  Motion.  —  The  last 
great  observer  before  the  invention  of  the  telescope  was  the 
Danish  astronomer  Tycho 
Brahe  (1546-1601).  He  was 
an  energetic  and  most  pains- 
taking worker.  He  not  only 
catalogued  many  stars,  but 
he  also  observed  comets, 
proving  they  are  beyond  the 
earth's  atmosphere,  and  ob- 
tained an  almost  continuous 
record  for  many  years  of  the 
positions  and  motions  of  the 
sun,  moon,  and  planets. 

Tycho  Brahe's  successor 
was  his  pupil  Kepler  (1571- 
1630),  who  spent  more  than 
20  years  in  attempting  to  find 
from  the  observations  of  his  master  the  manner  in  which  the 
planets  actually  move.  The  results  of  an  enormous  amount 
of  calculation  on  his  part  are  contained  in  the  following  three 
laws  of  planetary  motions  : 

I.  Every  planet  moves  so  that  the  line  joining  it  to  the  sun 
sweeps  over  equal  areas  in  equal  intervals  of  time,  whatever 
their  length.     This  is  known  as  the  law  of  areas. 

II.  The  orbit  of  every  planet  is  an  ellipse  with  the  sun  at 
one  of  its  foci. 


Johann  Kepler. 


230     AN  INTRODUCTION   TO  ASTRONOMY   [ch.  viii,  145 

III.  The  squares  of  the  periods  of  any  two  planets  are 
proportional  to  the  cubes  of  their  mean  distances  from  the 
sun. 

All  the  complexities  of  the  apparent  motions  of  the  planets 
are  explained  by  Kepler's  three  simple  laws  when  taken  in 
connection  with  the  periods  of  the  planets  and  the  positions 
of  their  orbits. 

146.  The  Law  of  Gravitation.  —  Newton  based  his  great- 
est discovery,  the  law  of  gravitation,  on  Kepler's  laws.  From 
each  one  of  them  he  drew  an  important  conclusion. 

Newton  proved  by  a  suitable  mathematical  discussion, 
based  on  his  laws  of  motion,  that  it  follows  from  Kepler's 
first  law  that  every  planet  is  acted  on  by  a  force  which  is  di- 
rected toward  the  sun.  This  was  the  first  time  that  the  sun 
and  planets  were  shown  to  be  connected  dynamically.  Be- 
fore Newton's  time  it  was  generally  supposed  that  there 
was  some  force  acting  on  the  planets  in  the  direction  of  their 
motion  which  kept  them  going  in  their  orbits. 

The  first  law  of  Kepler  led  to  the  conclusion  that  the  planets 
are  acted  on  by  forces  directed  toward  the  sun,  but  gave  no 
information  whatever  regarding  the  manner  in  which  the 
forces  depend  upon  the  position  of  the  planet.  The  second 
law  furnishes  a  basis  for  the  answer  to  this  question,  and 
from  it  Newton  proved  that  the  force  acting  on  each  planet 
varies  inversely  as  the  square  of  its  distance  from  the  sun. 

The  law  of  the  inverse  squares  is  encountered  in  many 
phenomena  besides  gravitation.  For  example,  it  holds  for 
magnetic'  and  electric  forces,  the  intensity  of  light  and  of 
sound,  and  the  magnitudes  of  water  and  earthquake  waves. 
The  reason  it  holds  for  the  radiation  of  light  is  easily  under- 
stood. The  area  of  the  spherical  surface  which  the  rays 
cross  in  proceeding  from  a  point  is  proportional  to  the 
square  of  its  radius.  Since  the  intensity  of  illumination  is 
inversely  proportional  to  the  illuminated  area,  it  is  inversely 
as  the  square  of  the  distance.  If  gravitation  in  some  way 
depended  on  lines  of  force  extending  out  from  matter  radially, 


CH.  viii,  147]  THE    SOLAR  SYSTEM  231 

it  would  vary  inversely  as  the  square  of  the  distance,  but 
nothing  is  positively  known  as  to  its  nature. 

Another  interesting  question  remains,  and  that  is  whether 
the  gravitation  of  a  body  is  strictly  proportional  to  its 
inertia,  regardless  of  its  constitution  and  condition,  or 
whether  it  depends  upon  its  composition,  temperature,  and 
other  characteristics.  All  other  known  forces,  such  as  mag- 
netism, depend  upon  other  things  than  mass,  and  it  might 
be  expected  the  same  would  be  true  of  gravitation.  But  it 
follows  from  Kepler's  third  law  that  the  sun's  attraction  for 
the  several  planets  is  independent  of  their  different  consti- 
tutions, motions,  and  physical  conditions.  Since  the  same 
law  holds  for  the  800  planetoids  as  well,  in  which  there  is 
opportunity  for  great  diversities,  it  is  concluded  that  gravita- 
tion depends  upon  nothing  whatever  except  the  masses  and 
the  distances  of  the  attracting  bodies. 

Suppose  the  attraction  between  unit  masses  at  unit  dis- 
tance is  taken  as  unity,  and  consider  the  attraction  of  a 
body  composed  of  many  units  for  another  of  many  units. 
To  fix  the  ideas,  suppose  one  body  has  5  units  of  mass  and 
the  other  4  imits;  the  problem  is  to  find  the  number  of 
units  of  force  between  them  at  distance  unity.  Each  of  the 
5  tmits  exerts  a  unit  of  force  on  each  of  the  4  units.  That 
is,  each  of  the  5  units  exerts  all  together  4  units  of  force  on 
the  second  body.  Therefore,  the  entire  first  body  exerts 
5  X  4  =  20  units  of  force  on  the  second  body ;  or,  the 
whole  force  is  proportional  to  the  products  of  the  masses. 

On  uniting  the  results  obtained  from  Kepler's  three  laws 
and  assuming  that  they  hold  always  and  everywhere,  the 
universal  law  of  gravitation  is  obtained : 

Every  particle  of  matter  in  the  universe  attracts  every  other 
particle  with  a  force  which  is  proportional  to  the  product  of 
their  masses,  and  which  varies  inversely  as  the  square  of  the 
distance  between  them. 

147.  The  Importance  of  the  Law  of  Gravitation.  —  The 
importance  of  a  physical  law  depends  upon  the  number  of 


232    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  viii,  147 

phen6mena.it  coordinates  and  upon  the  power  it  gives  the 
scientist  of  making  predictions.  Consider  the  law  of  gravi- 
tation in  these  respects.     In  his  great   work,    Philosophies 


Fig.  87.  —  Isaac  Newton. 


Naturalis  Principia  Mathematica  (The  Mathematical  Prin- 
ciples of  Natural  Philosophy),  commonly  called  simply  the 
Principia,  Newton  showed  how  every  known  phenomenon 
of  the  motions,  shapes,  and  tides  of  the  solar  system  could  be 
explained  by  the  law  of  gravitation.     That  is,  the  elliptical 


CH.  VIII,  147]  THE   SOLAR   SYSTEM  233 

paths  of  the  planets  and  the  moon,  the  slow  changes  in  their 
orbits  produced  by  their  shght  mutual  attractions,  the  oblate- 
ness  of  rotating  bodies,  the  precession  of  the  equinoxes,  and 
the  countless  small  irregularities  in  planetary  and  satelhte 
motions  that  can  be  detected  by  powerful  telescopes,  are 
all  harmonious  under  the  law  of  gravitation,  and  what  once 
seemed  to  be  a  hopeless  tangle  has  been  found  to  be  an 
orderly  system.  All  the  discoveries  in  this  direction  for  more 
than  200  years  have  confirmed  the  exactness  of  the  law 
of  gravitation  until  it  is  now  by  far  the  most  certainly 
established  physical  law. 

Not  only  is  the  law  of  gravitation  operative  in  the  great 
phenomena  where  its  effects  are  easy  to  detect,  but  also  in 
everything  in  which  the  motion  of  matter  is  involved.  It  is 
found  on  reflection  that  all  phenomena  depend  either  directly 
or  indirectly  upon  the  motion  of  matter,  for  even  changes 
of  the  mental  state  of  an  individual  are  accompanied  by 
corresponding  changes  in  the  structure  of  his  brain.  When 
a  petson  moves,  his  changed  relation  to  the  remainder  of 
the  universe  causes  a  corresponding  change  in  the  gravita- 
tional stress  by  which  he  is  connected  with  it ;  indeed,  when 
he  thinks,  the  alterations  in  his  brain  at  once  cause  alter- 
ations in  the  gravitational  forces  between  it  and  matter 
even  in  the  remotest  parts  of  space.  These  effects  are  cer- 
tainly real,  though  there  is  no  known  means  of  detecting 
them. 

The  law  of  gravitation  became  in  the  hands  of  the  suc- 
cessors of  Newton  one  of  the  most  valuable  means  of  dis- 
covery. Time  after  time  such  great  mathematicians  as 
Laplace  and  Lagrange,  using  it  as  a  basis,  predicted  things 
which  had  not  then  been  observed,  but  which  invariably 
were  found  later  to  be  true.  But  scientific  men  are  not 
contented  with  simply  making  predictions  and  finding  that 
they  come  true.  On  the  basis  of  their  established  laws  they 
seek  to  foresee  what  will'  happen  in  the  almost  indefinite 
future,  even  beyond  the  time  when  the  human  race  shall 


234    AN   INTRODUCTION   TO  ASTRONOMY  [ch.  viii,  147 

have  become  extinct,  and,  similarly,  what  the  conditions  were 
back  before  the  time  when  life  on  the  earth  began. 

The  law  of  gravitation  was  undoubtedly  Newton's  greatest 
discovery,  and  the  importance  of  it  and  his  other  scientific 
work  is  indicated  by  the  statements  of  competent  judges. 
The  brilliant  German  scholar,  Leibnitz  (I646-I716),  a  con- 
temporary of  Newton  and  his  greatest  rival,  said,  "  Taking 
mathematics  from  the  beginning  of  the  world  to  the  time 
when  Newton  hved,  what  he  had  done  was  much  the  better 
half."  The  French  mathematician,  Lagrange  (I736-18I3), 
one  of  the  greatest  masters  of  celestial  mechanics,  wrote, 
"  Newton  was  the  greatest  genius  that  ever  existed,  and  the 
most  fortunate,  for  we  cannot  find  more  than  once  a  sys- 
tem of  the  world  to  establish."  The  English  writer  on  the 
history  of  science,  Whewell,  said,  "  It  [the  law  of  gravita- 
tion] is  indisputably  and  incomparably  the  greatest  scientific 
discovery  ever  made,  whether  we  look  at  the  advance  which 
it  involved,  the  extent  of  the  truth  disclosed,  or  the  funda- 
mental and  satisfactory  nature  of  this  truth."  Compare 
these  splendid  and  deserved  eulogies  with  Newton's  own 
estimate  of  his  efforts  to  find  the  truth :  "  I  do  not  know 
what  I  may  appear  to  the  world ;  but  to  myself  I  seem  to 
have  been  only  like  a  boy  playing  on  the  seashore,  and 
diverting  myself  in  now  and  then  finding  a  smoother  pebble 
or  a  prettier  shell  than  ordinary,  while  the  great  ocean  of 
truth  lay  all  undiscovered  before  me."  There  is  every 
reason  to  believe  that  this  is  the  sincere  and  unaffected  ex- 
pression of  a  great  mind  which  realized  the  magnitude  of 
the  unknown  as  compared  to  the  known. 

In  Westminster  Abbey,  in  London,  Newton  lies  buried 
among  the  noblest  and  the  greatest  English  dead,  and  over 
his  tomb  on  a  tablet  they  have  justly  engraved,  "  Mortals, 
congratulate  yourselves  that  so  great  a  man  has  lived  for 
the  honor  of  the  human  race." 

148.  The  Conic  Sections.  —  After  having  found  that,  if 
the  orbit  of  a  body  is  an  ellipse  with  the  center  of  force  at  a 


CH.  viii,  148] 


THE   SOLAR  SYSTEM 


235 


.  focus,  then  the  force  to  which  it  is  subject  varies  inversely  as 

the  square  of  its  distance,  Newton  took  up  the  converse 

problem.     Under  the  assumption  that  the  attractive  force 

varies  inversely  as  the  square  of  the  distance,  he  proved 

that  the  orbit  must  be  what  is  called  a 

conic  section,  an  example  of  which  is  the 

ellipse. 

The  conic  sections  are  highly  interesting 

curves  first  studied  by  the  ancient  Greeks. 

They  derive  their  name  from  the  fact  that 

they  can  be  obtained  by  cutting  a  circular 

cone  with  planes.     In  Fig.  88  is  shown  a 

double  circular  cone  whose  vertex  is  at  V. 

A  plane  section  perpendicular  to  the  axis 

of  the  cone  gives  a  circle  C.     An  oblique 

section  gives  an  ellipse  E ;  however,  the 

plane  must  cut  both  sides  of  the  cone. 

When  the  plane  is  parallel  to  one  side,  or 

element,  of  the  cone,  a  parabola  P  is  ob- 
tained.    When  the  plane  cuts  the  two 

branches  of  the  double  cone,  the  two 
branches  of  an  hyperbola  HH  are  ob- 
tained. There  are  in  addition  to  these 
figures  certain  Umiting  cases.  One  is  that 
in  which  the  intersecting  plane  passes 
only  through  the  vertex  V  giving  a 
simple  point ;  another  is  that  in  which  the  intersecting  plane 
touches  only  one  element  of  the  cone,  giving  a  single  straight 
line;  and  the  last  is  that  in  which  the  intersecting  plane 
passes  through  the  vertex  B  and  cuts  both  branches  of  the 
cone,  giving  two  intersecting  straight  lines. 

The  character  of  the  conic  described  depends  entirely 
upon  the  central  force  and  the  way  in  which  the  body  is 
started.  For  example,  suppose  a  body  is  started  from  0, 
Fig.  89,  in  the  direction  OT,  perpendicular  to  OS.  If 
the  initial  velocity  of  the  body  is  zero,  it  will  fall  straight  to 


Fig.  88.  —  The  conic 
sections. 


236    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  viii,  148 


S.  If  the  initial  velocity  is  not  too  great,  it  will  describe  the 
elUpse  E,  and  0  will  be  the  aphehon  point.  If  the  initial 
velocity  is  just  great  enough  so  that  the  centrifugal  acceler- 
ation balances  the  attraction,  the  orbit  will  be  the  circle  C. 
If  the  initial  velocity  is  a  little  greater  than  that  in  the  circle, 
the  body  will  describe  the  eUipse  E',  and  0  will  be  the  peri- 
hehon  point.     If  the  initial  velocity  is   exactly  V2   times 

that  for  the  circular  orbit, 
the  body  will  move  in  the 
parabola  P.  If  the  initial 
velocity  is  still  greater,  the 
orbit  will  be  an  hyperbola  H. 
And  finally,  if  the  initial  ve- 
locity is  infinite,  the  path  will 
be  the  straight  line  whose 
direction  is  OT.  If  the  ini- 
tial direction  of  motion  is 
not  perpendicular  to  OS,  the 
results  are  analogous,  except 
that  there  is  then  no  initial 
velocity  which  will  give  a 
circular  orbit. 

It  is  seen  from  this  discus- 
sion that  it  is  as  natural  for  a 
body  to  move  in  one  conic 
section  as  in  another.  Some  of  the  satellites  move  in  orbits 
which  are  very  nearly  circular ;  the  planets  move  in  elUpses 
with  varjdng  degrees  of  elongation ;  many  comets  move  in 
orbits  which  are  sensible  parabolas ;  and  there  may  possibly 
be  comets  which  move  in  hyperbolas. 

149.  The  Question  of  other  Laws  of  Force.  —  Many 
other  laws  of  force  than  that  of  the  inverse  squares  are 
conceivable.  For  example,  the  intensity  of  a  force  might 
vary  inversely  as  the  third  power  of  the  distance.  The  char- 
acter of  the  curve  described  by  a  body  moving  subject  to  any 
such  force  can  be  determined  by  mathematical  processes. 


Fig.  89.  —  Different  conies  depending 
on  the  initial  velocity. 


CH.  VIII,  150]  THE    SOLAR   SYSTEM  237 


er 


It  is  found  that,  if  the  force  varied  according  to  any  oth„ 
power  of  the  distance  than  the  inverse  square,  except  directly 
as  the  first  power,  then  (save  in  special  initial  conditions)  the 
orbits  would  be  curves  leading  either  into  the  center  of  force 
or  out  to  infinity.  Such  a  law  would  of  course  be  fatal  to 
the  permanence  of  the  planetary  system. 

If  the  force  varied  directly  as  the  distance,  the  orbits 
would  all  be  exactly  ellipses,  in  spite  of  the  mutual  attrac- 
tions of  the  planets,  the  sun  would  be  at  the  center  of  all  the 
orbits,  and  all  the  periods  would  be  the  same.  This  would 
imply  an  enormous  speed  for  the  remote  bodies. 

150.  Perturbations.  —  If  the  planets  were  subject  to  no 
forces  except  the  attraction  of  the  sun,  their  orbits  would  be 
strictly  ellipses.  But,  according  to  the  law  of  gravitation, 
every  planet  attracts  every  other  planet.  Their  mutual 
attractions  are  small  compared  to  that  of  the  sun  because 
of  their  relatively  small  masses,  but  they  cause  sensible, 
though  small,  deviations  from  strict  elUptical  motion,  which 
are  called  perturbations. 

The  mutual  perturbations  of  the  planets  are  sometimes 
regarded  as  blemishes  on  what  would  be  otherwise  a  perfect 
system.  Such  a  point  of  view  is  quite  unjustified..  Each 
body  is  subject  to  certain  forces,  and  its  motion  is  the  result 
of  its  initial  position  and  velocity  and  these  forces.  If  the 
masses  of  the  planets  were  not  so  small  compared  to  that  of 
the  sun,  their  orbits  would  not  even  resemble  ellipses. 

The  problems  of  the  mutual  perturbations  of  the  planets 
and  those  of  the  perturbations  of  the  moon  are  exceedingly 
difiicult,  and  have  taxed  to  the  utmost  the  powers  of  .mathe- 
maticians. In  order  to  obtain  some  idea  of  their  nature  con- 
sider the  case  of  only  two  planets,  Pi  and  Pj-  The  forces 
that  Pi  and  Pa  would  exert  upon  each  other  if  they  both 
moved  in  their  unperturbed  elliptical  orbits  can  be  computed 
without  excessive  difficulty,  and  the  results  of  these  forces 
can  be  determined.  But  the  resulting  departures  from  ellip- 
tical motion  cause  corresponding  alterations  in  the  forces, 


238    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  viii,  150 

which  produce  new  perturbations.  These  new  perturbations 
in  turn  change  the  forces  again.  The  forces  give  rise  to  new 
perturbations,  and  the  perturbations  to  new  perturbing 
forces,  and  so  on  in  an  unending  sequence.  In  the  solar 
system  where  the  ,m3,sses  of  the  planets  are  small  compared 
to  that  of  the  sun,  the  perturbations  of  the  series  decrease 
very  rapidly  in  importance.  If  the  masses  of  the  planets 
were  large  compared  to  the  sun  so  that  Kepler's  laws  would 
not  have  been  even  approximately  true,  it  is  doubtful  if 
even  the  genius  of  Newton  could  have  extracted  from  the 
intricate  tangle  of  phenomena  the  master  principle  of  the 
celestial  motions,  the  law  of  gravitation. 

Although  the  perturbations  may  be  small,  the  question 
arises  whether  they  may  not  be  extremely  important  in  the 
long  run.  The  subject  was  treated  by  Lagrange  and  La- 
place toward  the  end  of  the  eighteenth  century.  They 
proved  that  the  mean  distances,  the  eccentricities,  and  the 
inclinations  of  the  planetary  orbits  oscillate  through  rela- 
tively narrow  ranges,  at  least  for  a  long  time.  If  these  re- 
sults were  not  true,  the  stability  of  the  system  would  be  im- 
periled, for  with  extreme  variation  of  especially  the  first  two 
of  these  quantities  the  characteristics  of  the  planetary  orbits 
would  be  entirely  changed.  On  the  other  hand,  the  peri- 
helion points  and  the  places  where  the  planes  of  the  orbits 
of  the  planets  intersect,  a  fixed  plane  not  only  have  small 
oscillations,  but  they  involve  terms  which  continually  change 
in  one  direction.  Examples  of  perturbations  of  precisely 
this  sort  already  encountered  are  the  precession  of  the  equi- 
noxes (Art.  47)  and  the  revolution  of  the  moon's  Une  of 
nodes  (Art.  119). 

151.  The  Discovery  of  Neptune.  —  Not  only  can  the  per- 
turbations be  computed  when  the  positions,  initial  motions, 
and  the  masses  of  the  planets  are  given,  but  the  converse 
problem  can  be  treated  with  some  success.  That  is,  if  the 
perturbations  are  furnished  by  the  observations,  the  nature 
of  the  forces  which  produce  them  can  be  inferred.     The  most 


CH.  VIII,  161] 


THE    SOLAR   SYSTEM 


239 


celebrated  example  of  this  converse  problem  led  to  the  dis- 
covery of  the  planet  Neptune. 

In  1781  William  Herschel  discovered  the  planet  Uranus 
while  carrying  out  his  project  of  examining  every  object  in 
the  heavens  within  reach  of  his  telescope.     After  it  had 
been  observed  for  some  time  its  orbit  was  computed.     In 
order  to  predict  its  position  exactly  it  was  necessary  to 
compute    the    perturba- 
tions  due  to  all  known 
bodies.      This  was  done 
by  Bouvard  on  the  basis 
of     the      mathematical 
theory  of  Laplace.     But 
by  1820  there  were  un- 
mistakable  discordances 
between  theory  and  ob- 
servation; by  1830,  they 
were  still  more  serious ; 
by  1840,  they  had  become 
intolerable.      This    does 
not  mean  that  prediction 
assigned    the    planet   to 
one  part  of  the  sky  and 
observation  found  it  in  a 
far  different  one ;  for,  in 
1840,  its  departure  from 
its     calculated    position 
amounted  to  only  two  thirds  the  apparent  distance  between 
the  two  components  of  Epsilon  Lyrse  (Art.  88),  a  quantity 
invisible  to  the  unaided  eye.     It  seems  incredible  that  so 
slight  a  discordance  between  theory  and  observation  after  60 
years  of  accumulation  could  have  led  to  any  valuable  results. 
By  1820  it  began  to  be  suggested  that  the  discrepancies 
in  the  motion  of  Uranus  might  be  due  to  the  attraction  of 
a  more  remote  unknown  planet.     The  problem  was  to  find 
the  unknown  planet.     Such  excessive  mathematical  difficul- 


FiG.  90. '—  William  Herschel. 


240    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  viii,  151 


FiQ.  91.  —  John  Couch  Adams. 


ties  were  involved  that  it  seemed  insoluble.  In  fact,  Sir 
George  Airy,  Astronomer  Royal  of  England,  expressed  him- 
self later  than  1840  as  not  be- 
lieving the  problem  could  be 
solved.  However,  a  yoimg 
Enghshman,  Adams,  and  a 
young  Frenchman,  Leyerrier, 
•  with  all  the  enthusiasm  of 
youth,  quite  independently  took 
up  the  problem  about  1845. 
Adams  finished  his  work  first 
and  communicated  his  results 
both  to  Challis,  at  Cambridge, 
and  to  Airy,  at  Greenwich. 
To  say  the  least,  they  took 
no  very  active  interest  in  the 
matter  and  allowed  the  search 
for  the  supposed  body  to  be 
postponed.  Adams  continued 
his  work  and  made  five  separate 
and  very  laborious  computa- 
tions. In  the  meantime  Le- 
verrier  completed  his  work  and 
sent  the  results  to  a  young 
German  astronomer,  Galle. 
Impatiently  Galle  waited  for 
the  night  and  the  stars.  On 
the  first  evening  after  receiv- 
ing Leverrier's  letter,  Septem- 
ber 23,  1846,  he  looked  for 
the  unknown  body,  and  found 
it  within  half  a  degree  of 
the  position  assigned  to  it 
by  Leverrier,  which  agreed 
substantially  with  that  indicated  by  Adams. 

Neptune  is  nearly  three  thousand  milhons  of  miles  from  the 


Fig.  92.  —  Joseph  Leverrier. 


CH.  viii,  152]  THE   SOLAR    SYSTEM  241 

earth,  beyond  the  reach  of  all  our  senses  except  that  of  sight, 
and  it  can  be  seen  only  with  telescopic  aid ;  its  distance  is 
so  great  that  more  than  four  hours  are  required  for  its  hght 
to  come  to  us,  yet  it  is  bound  to  the  remainder  of  the  sys- 
tem by  the  invisible  bonds  of  gravitation.  But  its  attrac- 
tion slightly  influenced  the  motions  of  Uranus,  and  from 
these  shght  disturbances  its  existence  and  position  were 
inferred.  Notwithstanding  the  fact  that  both  Adams  and 
Leverrier  made  assumptions  respecting  the  distance  of  the 
unknown  body  which  were  son^ewhat  in  error,  their  work 
stands  as  a  monument  to  the  reasoning  powers  of  the  human 
mind,  and  to  the  perfection  of  the  theory  of  the  motions  of 
the  heavenly  bodies. 

152.  The  Problem  of  Three  Bodies.  —  While  the  prob- 
lem of  two  mutually  attracting  bodies  presents  no  serious 
mathematical  troubles,  because  the  motion  is  always  in  some 
kind  of  a  conic  section,  that  of  three  bodies  is  one  of  the 
most  formidable  difliculty.  It  is  often  supposed  that  it  has 
not  been,  and  perhaps  that  it  cannot  be,  solved.  Such  an 
idea  is  incorrect,  as  will  now  be  explained. 

The  theory  of  the  perturbations  of  the  planets  is  really  a 
problem  of  three,  or  rather  of  eight,  bodies,  and  has  been 
completely  solved  for  an  interval  of  time  not  too  great.  That 
is,  while  the  orbits  of  the  bodies  cannot  be  described  for  an 
indefinite  interval  of  time  because  they  are  not  closed  curves 
but  wind  about  in  a  very  complicated  fashion,  nevertheless 
it  is  possible  to  compute  their  positions  with  any  desired 
degree  of  precision  for  any  time  not  too  remote.  There- 
fore, in  a  perfectly  real  and  just  sense  the  problem  has  been 
solved. 

There  are  particular  solutions  of  the  problem  of  three 
bodies  in  which  the  motion  can  be  described  for  any  period 
of  time,  however  long.  The  first  of  these  were  discovered 
by  Lagrange,  who  found  two  special  cases.  In  one  of  them 
the  bodies  move  so  as  to  remain  always  in  a  straight  line, 
and  in  the  other  so  as  to  be  always  at  the  vertices  of  an  equi- 


242    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  vm,  152 

lateral  triangle.  In  both  cases  the  orbits  are  conic  sections. 
In  1878  an  American  astronomer,  Hill,  in  connection  with 
his  work  on  the  motion  of  the  moon,  discovered  some  less 
simple  but  immensely  more  important  special  cases.  Since 
1890  Poincare,  universally  regarded  as  the  greatest  mathe- 
matician of  recent  times,  has  proved  the  existence  of  an 
infinite  number  of  these  special  cases  called  periodic  solutions. 
In  all  of  them  the  problem  is  exactly  solved.  Still  more 
recently  Sundman,  of  Helsingfors,  Finland,  has  in  an  im- 
portant mathematical  sense  solved  the  general  case.  How- 
ever, in  spite  of  all  the  results  that  have  been  achieved,  the 
problem  still  presents  to  the  mathematician  unsolved  ques- 
tions of  almost  infinite  variety. 

153.  The  Cause  of  the  Tides.  —  So  far  in  the  present 
discussion  only  the  effect  of  one  body  on  the  motion  of 
another,  taken  as  a  whole,  has  been  considered.  There 
remains  to  be  considered  the  distortion  of  one  body  by 
the  attraction  of  another.  These  deformations  give  rise  to 
the  tides. 

Before  proceeding  to  a  direct  discussion  of  the  tidal  prob- 
lem it  is  necessary  to  state  an  important  principle,  namely, 
if  two  bodies  are  subject  to  equal  parallel  accelerations,  their 
relative  positions  are  not  changed.  The  truth  of  this  propo- 
sition follows  from  the  laws  of  motion,  but  it  is  better  un- 
derstood from  an  illustration.  Suppose  two  bodies  of  the 
same  or  different  dimensions  are  dropped  from  the  top  of  a 
high  tower.  They  have  initially  a  certain  relation  to  each 
other  and  they  are  subject  to  equal  parallel  accelerations, 
namely,  those  produced  by  the  earth's  attraction.  In  their 
descent  they  fall  faster  and  faster ;  but,  neglecting  the  effects 
of  the  resistance  of  the  air,  they  preserve  the  same  relations 
to  each  other. 

Let  E,  Fig.  93,  represent  the  earth,  and  0  and  0'  two 
points  on  its  surface.  Consider  the  tendency  of  the  moon 
M  to  displace  0  on  the  surface  of  the  earth.  The  moon  at- 
tracts the  center  of  the  earth  E  in  the  direction  EM.    Let 


CH.  VIII,  153] 


THE    SOLAR   SYSTEM 


243 


its  acceleration  be  represented  by  EP.  In  the  same  units 
OA  represents  the  acceleration  of  Af  on  0  in  direction  and 
amount.  The  line  OA  'is  greater  than  EP  because  the 
moon  is  nearer  to  .0  than  it  is  to  E.  Now  resolve  OA  into 
two  components,  one  of  which,  OB,  shall  be  equal  and  par- 
allel to  EP.  The  other  component  is  OC.  Since  OB  and  EP 
are  equal  and  parallel,  it  follows  from  the  principle  stated 


-*fl' 


Fig.  93.  —  Resolution  of  the  tide-raising  forces. 


at  the  beginning  of  this  article  that  they  do  not  change  the 
relative  positions  of  E  and  0.  Therefore  OC,  the  outstand- 
ing component,  represents  the  tide-raising  acceleration  both 
in  direction  and  amount. 

The  results  for  0'  are  analogous,  and  the  tide-raising 
force  O'C  is  directed  away  from  the  moon  because  O'A'  is 
shorter  than   EP.     Figure  94  shows  the  tide-raising  ac- 


Fig.  94.  —  The  tide-raising  forces. 


celerations  around  the  whole  circumference  of  .the  earth. 
This  method  of  deriving  the  tide-raising  forces  is  the  ele- 
mentary geometrical  counterpart  of  the  rigorous  mathe- 


244  AN   INTRODUCTION   TO   ASTRONOMY     [ch.  vni,  163 

matical  treatment/  and  it  can  be  relied  on  to  give  correctly 
all  that  there  is  in  this  part  of  the  subject. 

A  more  detailed  discussion  than  can  be  entered  into  here 
shows  that  the  tide-raising  forces  are  about  5  per  cent 
greater  on  the  side  of  the  earth  which  is  toward  the  moon 
than  on  the  side  away  from  the  moon.  The  forces  outward 
from  the  surface  of  the  earth  in  the  line  of  the  moon  are 
about  twice  as  great  as  those  which  are  directed  inward  90° 
from  this  line.  The  tidal  forces  due  to  the  sun  are  a  little 
less  than  half  as  great  as  those  due  to  the  moon ;  no  other 
bodies  have  sensible  tidal  effects  on  the  earth. 

154.  The  Masses  of  Celestial  Bodies.  —  The  masses  of 
celestial  bodies  are  determined  from  their  attractions  for 
other  bodies.  Suppose  a  satellite  revolves  around  a  planet 
in  an  orbit  of  measured  dimensions  in  an  observed  period. 
From  these  data  it  is  possible  to  compute  the  acceleration  of 
the  planet  for  the  satellite  because  the  attraction  balances 
the  centrifugal  acceleration.  It  is  possible  to  determine 
what  the  earth's  attraction  would  be  at  the  same  distance, 
and,  consequently,  the  relation  of  its  mass  to  that  of  the 
other  planet.  There  has  been  much  difficulty  in  finding 
the  masses  of  Mercury  and  Venus  because  they  have  no 
known  satellites.  Their  masses  have  been  determined  with 
considerable  reliability  from  their  perturbations  of  each 
other  and  of  the  earth,  and  from  their  perturbations  of  cer- 
tain comets  that  have  passed  near  them. 

A  useful  formula  for  the  sum  of  the  masses  of  any  two 
bodies  mi  and  nii  which  attract  each  other  according  to  the 
law  of  gravitation,  for  example,  the  two  components  of  a 
double  star,  is 

where  a  is  the  distance  between  the  bodies  expressed  in 

'  An  analytical  discussion  proves  that  the  tide-raising  force. is  propor- 
tional to  the  product  of  the  mass  of  the  disturbing  body  and  the  radius  of 
the  disturbed  body,  and  inversely  proportional  to  the  cube  of  the  distance 
between  the  disturbing  and  disturbed  bodies. 


CH.  VIII,  155]  THE    SOLAR   SYSTEM  245 

terms  of  the  earth's  distance  from  the  sun  as  unity,  and 
where  P  is  the  period  expressed  in  years.  The  sum  of  the 
masses  is  expressed  in  terms  of  the  sun's  mass  as  unity. 

155.  The  Surface  Gravity  of  Celestial  Bodies.  —  The 
surface  gravity  of  a  celestial  body  is  an  important  factor  in 
the  determination  of  its  surface  conditions,  and  is  funda- 
mental in  the  question  of  its  retaining  an  atmosphere.  The 
surface  gravity  of  a  spherical  body  depends  only  upon  its 
mass  and  dimensions. 

Let  m  represent  the  mass  of  the  earth,  g  its  surface  gravity, 
,  and  r  its  radius.     Then  by  the  law  of  gravitation 

where  k^  is  a  constant  depending  on  the  units  employed. 
Let  M,  G,  and  R  represent  in  the  same  units  the  correspond- 
ing quantities  for  another  body.     Then 

R^ 
On  dividing  the  second  equation  by  the  first,  it  is  found  that 

GmA\ 

g  m  \RJ 
from  which  the  surface  gravity  G  can  be  found  in  terms  of 
that  of  the  earth  when  the  mass  and  radius  of  M  are  given. 
It  is  sometimes  convenient  to  have  the  expression  for  the 
ratio  of  the  gravities  of  two  bodies  in  terms  of  their  densities 
and  dimensions.  Let  d  and  D  represent  the  densities  of 
the  earth  and  the  other  body  respectively.  Then,  since 
m  =  f  Trdr*  and.M  =  |-7rZ)i?^'it  is  found  that 

G^  DR_ 
g  d  r 
That  is,  the  surface  gravities  of  celestial  bodies  are  pro- 
portional to  the  products  of  their  densities  and  radii.  A 
small  density  may  be  more  than  counterbalanced  by  a  large 
radius,  as,  for  example,  in  the  case  of  the  sun,  whose  density 
is  only  one  fourth  that  of  the  earth  but  whose  surface  gravity 
is  about  27.6  times  that  of  the  earth. 


246    AN  INTRODUCTION  TO  ASTRONOMY      [ch.  viii,  155 

XI.  QUESTIONS 

1.  If  the  sidereal  period  of  a  planet  were  half  that  of  the  earth, 
what  would  be  its  period  from  greatest  eastern  elongation  to  its  next 
succeeding  greatest  eastern  elongation  ? 

2.  If  the  sidereal  period  of  a  planet  were  twice  that  of  the  earth, 
what  would  be  its  period  from  opposition  to  its  next  succeeding 
opposition  ? 

3.  What  would  be  the  period  of  a  planet  if  its  mean  distance  from 
the  sun  were  twice  that  of  the  earth  ? 

4.  What  would  be  the  mean  distance  of  a  planet  if  its  period  were 
twice  that  of  the  earth  ? 

5.  The  motion  of  the  moon  around  the  earth  satisfies  (nearly) 
Kepler's  first  two  laws.  What  are  the  respective  conclusions  which 
follow  from,  them  ? 

6.  The  force  of  gravitation  varies  directly  as  the  product  of  the 
masses.  Show  that  the  acceleration  of  one  body  with  respect  to 
another,  both  being  free  to  move,  is  proportional  to  the  sum  of 
their  masses.     Hint.     Use  both  the  second  and  third  laws  of  motion. 

7.  In  Lagrange's  two  special  solutions  of  the  problem  of  three 
bodies  the  law  of  areas  is  satisfied  for  each  body  separately  with 
respect  to  the  center  of  gravity  of  the  three.  What  conclusion 
follows  from  this  fact  ?  How  does  the  force  toward  the  center  of 
gravity  vary  ? 

II.     The  Orbits,  Dimensions,  and  Masses  of  the 
Planets 

156.   Finding  the  actual  Scale  of  the  Solar  System.  —  It 

was  seen  in  Art.  144  that  the  relative  dimensions  of  the 
solar  system  can  be  determined  without  knowing  any  actual 
distance.  It  follows  from  this  that  if  the  distance  between 
any  two  bodies  can  be  found,  all  the  other  distances  can  be 
computed. 

The  problem  of  finding  the  actual  scale  of  the  solar  system 
is  of  great  importance,  because  the  determination  of  the 
dimensions  of  all  its  members  depends  upon  its  solution, 
and  the  distance  from  the  earth  to  the  sun  is  involved  in 
measuring  the  distances  to  the  stars.  Not  until  after  the 
year  1700  had  it  been  solved  with  any  considerable  degree 
of  approximation,  but  the  distance  from  the  earth  to  the  sun 


CH.  VIII,  156]  THE   SOLAR   SYSTEM  247 

is  now  known  with  an  error  probably  not  exceeding  one 
part  in  a  thousand. 

The  direct  method  of  measuring  the  distance  to  the  sun, 
analogous  to  that  used  in  case  of  the  moon  (Art.  123),  is  of 
no  value  because  the  apparent  displacement  to  be  measured 
is  very  small,  the  sun  is  a  body  with  no  permanent  surface 
markings,  and  its  heat  seriously  disturbs  the  instruments. 
But,  as  has  been  seen  (Art.  144),  the  distance  from  the  earth 
to  any  other  member  of  the  system  is  equally  useful,  and  in 
some  cases  the  measurement  of  the  distances  to  the  other 
bodies  is  feasible. 

Gill,  at  the  Cape  of  Good  Hope,  measured  the  distance 
of  Mars  with  considerable  success,  but  its  disk  and  red 
color  introduced  difficulties.  These  difficulties  do  not  arise 
in  the  case  of  the  smaller  planetoids,  which  appear  as  star- 
like points  of  light,  but  their  great  distances  decrease  the 
accuracy  of  the  results  by  reducing  the  magnitude  of  the 
quantity  to  be  measured.  However,  in  1898,  Witt,  of  Ber- 
Un,  discovered  a  planetoid  whose  orbit  lies  largely  within  the 
orbit  of  Mars  and  which  approaches  closer  to  the  earth  than 
any  other  celestial  body  save  the  moon.  Its  nearness,  its 
minuteness,  and  its  absence  of  marked  color  all  unite  to 
make  it  the  most  advantageous  known  body  for  getting  the 
scale  of  the  solar  system  by  the  direct  m€fthod.  Hinks,  of 
Cambridge,  England,  made  measurements  and  reductions  of 
photographs  secured  at  many  observatories,  and  found  that 
the  parallax  of  the  sun,  or  the  angle  subtended  by  the  earth's 
radius  at  the  mean  distance  of  the  sun,  is  8".8,  corresponding 
to  a  distance  of  92,897,000  miles  from  the  earth  to  the  sun. 

The  distance  of  the  earth  from  the  sun  can  also  be  found 
from  the  aberration  of  light.  The  amount  of  the  aberration 
depends  upon  the  velocity  of  light  and  the  speed  with  which 
the  observer  moves  across  the  line  of  its  rays.  The  velocity 
of  light  has  been  found  with  great  accuracy  from  experiments 
on  the  surface  of  the  earth.  The  amount  of  the  aberration 
has  been  determined  by  observations  of  the  stars.    From 


248  AN   INTRODUCTION   TO   ASTRONOMY    [ch.  viii,  156 

the  two  sets  of  data  the  velocity  of  the  observer  can  be 
computed.  Since  the  length  of  the  year  is  known,  the  length 
of  the  earth's  orbit  can  be  obtained.  Then  it  is  an  easy  matter, 
making  use  of  the  shape  of  the  orbit,  to  compute  the  mean 
distance  from  the  earth  to  the  sun.  The  results  obtained  in 
this  way  agree  with  those  furnished  by  the  direct  method. 

Another  and  closely  related  method  depends  upon  the 
determination  of  the  earth's  motion  in  the  line  of  sight 
(Art.  226)  by  means  of  the  spectroscope.  Spectroscopic 
technique  has  been  so  highly  perfected  that  when  stars  best 
suited  for  the  purpose  are  used  the  results  obtained  give  the 
earth's  speed  with  a  high  degree  of  accuracy.  Its  velocity 
and  period  furnish  the  distance  to  the  sun,  as  in  the  method 
depending  upon  the  aberration,  and  the  results  are  about  as 
accurate  as  those  furnished  by  any  other  method. 

There  are  several  other  methods  for  finding  the  distance 
to  the  sun  which  have  been  employed  with  more  or  less  suc- 
cess. One  of  them  depends  upon  transits  of  Venus  across 
the  sun's  disk.  Another  involves  the  attraction  of  the  sun 
for  the  moon.  But  none  of  them  is  so  accurate  as  those 
which  have  been  described. 

157.  The  Elements  of  the  Orbits  of  the  Planets.  —  The 
position  of  a  planet  at  any  time  depends  upon  the  size,  shape, 
and  position  of  its  orbit,  together  with  the  time  when  it  was 
at  some  particular  position,  as  the  perihelion  point.  These 
quantities  are  called  the  elements  of  an  orbit,  and  when  they 
are  given  it  is  possible  to  compute  the  position  of  the  planet 
at  any  time. 

The  size  of  an  orbit  is  determined  by  the  length  of  its  major 
axis.  It  is  an  interesting  and  important  fact  that  the  period 
of  revolution  of  a  planet  depends  only  upon  the  major  axis 
of  its  orbit,  and  not  upon  its  eccentricity  or  any  other  ele- 
ment. The  shape  of  an  orbit  is  defined  by  its  eccentricity. 
The  position  of  a  planet's  orbit  is  determined  by  its  orienta- 
tion in  its  plane  and  the  relation  of  its  plane  to  some  standard 
plane  of  reference.     The  longitude  of  the  perihelion  point 


CH.  viii,  157] 


THE    SOLAR   SYSTEM 


249 


defines  the  orientation  of  an  orbit  in  its  plane.  The  plane 
of  reference  in  common  use  is  the  plane  of  the  echptic.  The 
position  of  the  plane  of  the  orbit  is  defined  by  the  location 
of  the  Ime  of  its  intersection  with  the  plane  of  the  ecliptic 
and  the  angle  between  the  two  planes.  The  distance  from 
the  vernal  equinox  eastward  to  the  point  where  the  orbit  of 
the  body  crosses  the  echptic 
from  south  to  north  is  called 
the  longitude  of  the  ascend- 
ing node,  and  the  angle  be- 
tween the  plane  of  the  echp- 
tic and  the  plane  of  the  orbit 
is  called  the  inclination. 

In  Fig.  95,  VNQ  represents 
the  plane  of  the  ecliptic  and 
SNP  the  plane  of  the  orbit. 
The  vernal  equinox  is  at  V, 
the  angle  VSN  is  the  longi- 
tude of  the  ascending  node, 
the  angle  VSN  -|-  NSP  is 
the  longitude  of  the  perihelion,  and  the  angle  QNP  is  the 
inclination  of  the  orbit. 

The  elements  of  the  orbits  of  the  planets,  which  change 
very  slowly,  are  given  for  January  1,  1916,  in  Table  IV. 

Table  IV 


Fig 


95.  —  Elements  of  the  orbit  of 
a  planet. 


Planet 

Dis- 
tance, 
Mil- 
lions OF 

Miles 

Period 

IN 

Years 

Eccen- 
tricity 

Incli- 
nation 

TO 

Eclip- 
tic 

Longi- 
tude OF 
Node 

Longi- 
tude OF 
Perihe- 
lion 

Longi- 
tude ON 
Jan.  ], 
1916 

Mercury 

36.0 

0.241 

0.20562 

7°     0' 

47°  20' 

76°     9' 

334°     2' 

Venus      . 

67.2 

0.615 

0.00681 

3     24 

75    55 

130     23 

345    50 

Earth      . 

92.9 

1.000 

0.01674 

0    00 

101     30 

99    49 

Mars .     . 

141.5 

1.881 

0.09332 

1     51 

48    55 

334    31 

116    25 

Jupiter   . 

483.3 

11.862 

0  04836 

1     18 

99    36 

12    58 

3    51 

Saturn    . 

886.0 

29.458 

0  05583 

2    30 

112    55 

91    24 

102    20 

Uranus    . 

1781.9 

84.015 

0  04709 

0    46 

73    34 

169    18 

312       9 

Neptune 

2791.6 

164.788 

0.00854 

1     47 

130    51 

43    54 

120     12 

250    AN   INTRODUCTION   TO   ASTRONOMY  [ch.  viii,  157 

To  the  elements  of  the  orbits  of  the  planets  must  be 
added  the  direction  of  their  motion  in  order  to  be  altogether 
complete.  The  result  is  very  simple,  for  they  all  revolve  in 
the  same  direction,  namely,  eastward. 

The  most  interesting  and  important  element  of  the  plane- 
tary orbits  is  the  mean  distance.  The  distance  of  Neptune 
from  the  sun  is  30  times  that  of  the  earth  and  nearly  80 
times  that  of  Mercury.  Since  the  amount  of  light  and 
heat  received  per  unit  area  by  a  planet  varies  inversely  as 
the  square  of  its  distance  from  the  sun,  it  follows  that  if  the 
units  are  chosen  so  that  the  amount  received  by  the  earth 
is  unity,  then  the  respective  amounts  received  by  the  several 
planets  are:  Mercury,  6.66;  Venus,  1.91;  Earth,  1.00; 
Mars,  0.43;  Jupiter,  0.037 ;  Saturn,  0.011 ;  Uranus,  0.0027 ; 
Neptune,  0.0011.  It  is  seen  that  the  earth  receives  more 
than  900  times  as  much  light  and  heat  per  unit  area  as  Nep- 
tune, and  that  in  the  case  of  Mercury  and  Neptune  the 
ratio  is  more  than  6000.  Obviously,  other  things  being 
equal,  the  climatic  conditions  on  planets  differing  so  greatly 
in  distance  from  the  sun  would  be  enormous. 

As  seen  from  Neptune  the  sun  presents  a  smaller  disk 
than  Venus  does  to  us  when  nearest  to  the  earth.  It  is 
sometimes  supposed  that  Neptune  is  far  away  in  the  night 
of  space  where  the  sun  looks  simply  like  a  bright  star.  This 
is  far  from  the  truth,  for,  since  the  sunlight  received  by  the 
earth  is  600,000  times  full  moonhght,  and  Neptune  gets 
9^  as  much  light  as  the  earth,  it  follows  that  the  illu- 
mination of  Neptune  by  the  sun  is  nearly  700  times  that  of  the 
earth  by  the  brightest  full  moon.  Another  erroneous  idea 
frequently  held  is  that  Neptune  is  so  far  away  from  the  sun 
that  it  gets  a  considerable  fraction  of  its  Ught  from  other 
suns.  The  nearest  known  star  is  more  than  9000  times  as 
distant  from  Neptune  as  Neptune  is  from  the  sun,  and,  con- 
sequently, Neptune  receives  more  than  80,000,000  times  as 
much  light  and  heat  as  it  would  if  the  sun  were  at  the  dis- 
tance of  the  nearest  star. 


CH.  VIII,  157]  THE   SOLAR   SYSTEM  251 

It  is  almost  impossible  to  get  a  correct  mental  picture  of 
the  enormous  dimensions  of  the  solar  system,  and  there  are 
often  misconcaptions  in  regard  to  the  relative  dimensions  of 
the  orbits  of  the  various  planets.  To  assist  in  grasping  these 
distances,  suppose  one  has  traveled  sufficiently  to  have 
obtained  some  comprehension  of  the  great  size  of  the  earth. 
Then  he  is  in  a  position  to  attempt  to  appreciate  the  distance 
to  the  moon,  which  is  so  far  that  in  spite  of  the  fact  it  is  more 
than  2000  miles  in  diameter,  it  is  apparently  covered  by  a 
one-cent  piece  held  at  the  distance  of  6.5  feet.  In  terms 
of  the  earth's  dimensions,  its  distance  is  about  10  times  the 
circumference  of  the  earth.  It  is  so  remote  that  about  14 
days  would  be  required  for  sound  to  come  from  it  to  the 
earth  if  there  were  an  atmosphere  the  whole  distance  to  trans- 
mit it  at  the  rate  of  a  mile  in  5  seconds. 

Now  consider  the  distance  to  the  sun ;  it  is  400  times  that 
to  the  moon.  If  the  earth  and  sun  were  put  4  inches  apart 
on  such  a  diagram  as  could  be  printed  in  this  book,  on  the 
same  scale  the  distance  from  the  earth  to  the  moon  would  be 
Ym  of  3,n  inch.  If  sound  could  come  from  the  sun  to  the 
earth  with  the  speed  at  which  it  travels  in  air,  15  years  would 
be  required  for  it  to  cross  the  92,900,000  of  miles  between 
the  earth  and  sun.  Some  one,  having  found  out  at  what 
rate  sensations  travel  along  the  nerve  fibers  from  the  hand 
to  the  brain,  proved  by  calculation  that  if  a  small  boy  with 
a  sufficiently  long  arm  should  reach  out  to  the  sun  and  burn 
his  hand  off,  the  sensation  would  not  arrive  at  his  brain  so 
that  he  would  be  aware  of  his  loss  unless  he  lived  to  be  more 
than  100  years  of  age. 

The  relative  dimensions  of  the  orbits  of  the  planets  can  be 
best  understood  from  diagrams.  Unfortunately,  it  is  not 
possible  to  represent  them  to  scale  all  on  the  same  diagram. 
■Figure  96  shows  the  orbits  of  the  first  four  planets,  together 
with  that  of  Eros,  which  occupies  a  unique  position,  and  which 
has  been  used  in  getting  the  scale  of  the  system.  Figure  97 
shows  the  orbits  of  the  planets  from  Mars  to  Neptune  on  a 


252    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  viii,  157 

scale  which  is  about  -^  that  of  the  preceding  figure.     The 
most  noteworthy  fact  is  the  relative  nearness  of  the  four 


ORBiT  OF  ^MRs     , 


Fig.  96.  —  Orbits  of  the  four  inner  planets. 


inner  planets  and  the  enormous  distances  that  separate  the 
outer  ones. 

158.  The  Dimensions,  Masses,  and  Rotation  Periods  of 
the  Planets.  —  The  planetS'  Mercury  and  Venus  have  no 
known  satellites  and  their  masses  are  subject  to  some  un- 
certainties. The  rotation  periods  of  Mercury  and  Venus' 
are  very  much  in  doubt  because  of  their  unfavorable  po- 
sitions for  observation,  while  the  distances  of  Uranus  and 
Neptune  are  so  great  that  so  far  it  has  been  impossible  to 


CH.  VIII,  158]  THE    SOLAR   SYSTEM  253 

see  clearly  any  markings  on  their  surfaces.     There  is  some 
uncertainty  in  the  diameters  of  the  planets  on  account  of 


Fig.  97.  —  Orbit  of  the  outer  planets. 

what  is  called  irradiation,  which  makes  a  luminous  object 
appear  larger  than  it  actually  is. 

The  data  given  in  Table  V  are  based  partly  on  Barnard's 
many  measures  at  the  Lick  Observatory,  and  partly  on 
those  adopted  for  the  American  Ephemeris  and  Nautical 
Almanac. 


254    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  viii,  158 


Table  V 


Body 

Mean 

Mass 

Density 

Surface 

Gravity 

(9  =  1) 

Period  of 

Inclina- 
tion OF 

Diameter 

(Earth  =  1) 

(Water=1) 

Rotation 

Equator 
TO  Orbit 

Sun   .     . 

864,392 

329,390 

1.40 

27.64 

25  d.  8  h. 

7°  15' 

Moon     . 

2,160 

0.0122 

3.34 

0.16 

27  d.  7.7  h. 

6°  41' 

Mercury 

3,009 

0.045(?) 

4.48(  ?) 

0.31  (?) 

? 

? 

Venus    . 

7,701 

0.807  (?) 

4.85(?) 

0.85 

? 

? 

Earth     . 

7,918 

1.0000 

5.53 

1.00 

23  h.  56  m. 

23°  27' 

Mars      . 

4,339 

0.1065 

3.58 

0.36 

24  h.  37  m. 

23°  59' 

Jupiter  . 

88,392 

314.50 

1.25 

2.52 

9  h.  55  m. 

3° 

Saturn  . 

74,163 

94.07 

0.63 

1.07 

10  h.  14  m. 

27° 

Uranus  . 

30,193 

14.40 

1.44 

0.99 

? 

? 

Neptune 

34,823 

16.72 

1.09 

0.86 

? 

? 

Some  interesting  facts  are  revealed  by  this  table.  The 
first  four  planets  are  very  small  compared  to  the  outer  four, 
and  since  their  volumes  are  as  the  cubes  of  their  diameters, 


•     • 


Fig.  98.  —  Relative  dimensions  of  sun  and  planets. 

the  latter  average  more  than  a  thousand  times  greater  in 
volume  than  the  former.  The  inner  planets  are  much  denser 
than  the  outer  ones  and,  so  far  as  known,  rotate  on  their 
axes  more  slowly. 

Figure  98  shows  an  arc  of  the  sim's  circumference  and  the 


CH.  VIII,  159]  THE    SOLAR   SYSTEM  255 

eight  planets  to  the  same  scale.  It  is  apparent  from  this 
diagram  how  insignificant  the  earth  is  in  comparison  with 
the  larger  planets,  and  how  small  they  are  all  together  in 
comparison  with  the  sun. 

159.  The  Times  for  Observing  the  Planets.  —  Mercury 
and  Venus  are  most  conveniently  situated  for  observation 
when  they  are  near  their  greatest  elongations,  for  then  they 
are  not  dimmed  by  the  more  brilliant  rays  of  the  sun.  When 
they  are  east  of  the  sun  they  can  be  seen  in  the  evening,  and 
when  they  are  west  of  the  sun  they  are  observable  only  in 
the  morning.  Ordinarily  the  evening  is  more  convenient 
for  making  observations  than  the  morning,  and  therefore 
the  results  will  be  given  only  for  this  time. 

Those  planets  wh  ch  are  farther  from  the  sun  than  the 
earth  can  be  observed  best  when  they  are  in  opposition,  or 
180°  from  the  sun,  for  then  they  are  nearest  the  earth  and 
their  illuminated  sides  are  toward  the  earth.  When  a  planet 
is  in  opposition  it  crosses  the  meridian  at  midnight,  and  it 
can  be  observed  late  in  the  evening  in  the  eastern  or  south- 
eastern sky. 

The  problem  arises  of  determining  at  what  times  Mer- 
cury and  Venus  are  at  greatest  eastern  elongation,  and  at 
what  times  the  other  planets  are  in  opposition.  If  the  time 
at  which  a  planet  has  its  greatest  eastern  elongation  is  once 
given,  the  dates  of  all  succeeding  eastern  elongations  can 
be  obtained  by  adding  to  the  original  one  multiples  of  its 
synodical  period.  If  S  represents  the  synodical  period  of  an 
inferior  planet,  P  its  sidereal  period,  and  E  the  earth's  period, 
the  synodical  period  is  given  by  (Arts.  120,  144) 

1  =  1-1- 
S     P     E' 

and  in  the  case  of  a  superior  planet  the  corresponding  formula 

for  the  synodical  period  is 

1  =  1-1 
SEP 

On  the  basis  of  the  sidereal  periods  given  in  Table  IV,  these 


266    AN    INTRODUCTION   TO   ASTRONOMY   [ch.  viii,  159 

formulas,  and  data  from  the  American  Ephemeris  and  Nau- 
tical Almanac,  the  following  table  has  been  constructed:*. 


Table  VI 


Planet 

Eastern  Elonga- 
tion OR  Opposition 

Stnodical  Period 

Mercury  .     . 

Sept.    9,  1916 

Oyr. 

3  mo.  24.2  d.  =  0.31726  yr 

Venus  .     .     . 

April  23,  1916 

lyr. 

7  mo.    5.7  d.  =  1.59882  yr. 

Mars    .     .     . 

Feb.     9,  1916 

2yr. 

1  mo.  18.7  d.  =  2.13523  yr. 

Jupiter 

Oct.   23,  1916 

lyr. 

1  mo.    3.1  d.  =  1.09206  yr. 

Saturn      .     . 

Jan.      4,  1916 

lyr. 

0  mo.  12.6  d.  =  1.03514  yr. 

Uranus     .     . 

Aug.   10,  1916 

lyr. 

Omo.    4.3  d.  =  1.01205  yr. 

Neptune  . 

Jan.    22,  1916 

lyr. 

0  mo,    2.2  d.  =  1.00611  yr. 

The  superior  planets  are  most  brilliant  when  they  are 
in  opposition;  the  inferior  planets  are  brightest  some  time 
after  their  greatest  eastern  elongation  because  they  are 
then  relatively  approaching  the  earth  and  their  decrease  in 
distance  more  than  offsets  their  diminishing  phase.  For 
example,  in  1916  Venus  was  at  its  greatest  eastern  elongation 
April  23,  but  kept  getting  brighter  until  May  27. 

Mercury  is  so  much  nearer  the  sun  than  the  earth  that 
its  greatest  elongation  averages  only  23°,  though  it  varies 
from  18°  to  28°  because  of  the  eccentricity  of  the  orbit  of 
the  planet.  Consequently,  it  can  be  observed  only  for  a 
very  short  time  after  the  sun  is  far  enough  below  the  horizon 
for  the  brightest  stars  to  be  visible.  Mercury  at  its  brightest 
is  somewhat  brighter  than  a  first-magnitude  star.  There  is 
no  difficulty  in  observing  any  of  the  other  planets  except 
Uranus  and  Neptune,  Uranus  being  near  the  limits  of  visi- 
bility without  optical  aid,  and  Neptune  being  quite  beyond 
them.  Venus  is  brilliantly  white  and  at  its  brightest  quite 
surpasses  every  other  celestial  object  except  the  sun  and 
moon.     Mars  is  of  the  first  magnitude  and  decidedly  red. 


'  In  this  table  the  tropical  year  is  used  and  30  days  are  taken  as  constitut- 
ing a  month. 


CH.  VIII,  160]  THE   SOLAR   SYSTEM  257 

Jupiter  is  white  and  next  to  Venus  in  brilliance.  Saturn  is 
of  the  first  magnitude  and  slightly  yellowish. 

160.  The  Planetoids.  —  On  examination  it  is  found  that 
the  distance  of  each  planet  from  the  sun  is  roughly  twice 
that  of  the  preceding,  with  the  exception  of  Jupiter,  whose 
distance  is  about  3.5  times  that  of  Mars.  In  1772  Titius 
derived  a  series  of  numbers  by  a  simple  law  which  gave  the 
distances  of  the  planets  (Uranus  and.  Neptune  were  not 
known  then)  with  considerable  accuracy,  except  that  there 
was  a  number  for  the  vacant  space  between  Mars  and 
Jupiter.  The  law  is  that  if  4  is  added  to.each  of  the  num- 
bers 0,  3,  6,  12,  24,  48,  the  sums  thus  obtained  are  nearly 
proportional  to  the  distances  of  the  planets  from  the  sun. 
This  law,  commonly  called  Bode's  law,  because  the  writings 
of  Bode  made  it  widely  known,  rests  on  no  scientific  basis 
and  entirely  breaks  down  for  Neptune,  but  it  played  an 
important  role  in  two  discoyeries.  One  of  these  was  that 
both  Adams  and  Leverrier  assigned  distances  to  the  planet 
Neptune  on  the  basis  of  this  law,  and  computed  the  other 
elements  of  its  orbit  from  its  perturbations  of  Uranus  (Art. 
151).  The  other  discovery  to  which  Bode's  law  contributed 
was  that  of  the  planetoids. 

Toward  the  end  of  the  eighteenth  century  the  idea  became 
widespread  among  astronomers  that  there  was  probably  an 
undiscovered  planet  between  Mars  and  Jupiter  whose  dis- 
tance would  agree  with  the  fifth  number  of  the  Bode  series. 
In  1800  a  number  of  German  astronomers  laid  plans  to  search 
for  it,  but  before  their  work  was  actually  begun  Piazzi,  at 
Palermo,  on  January  1,  1801,  the  first  day  of  the  nineteenth 
century,  made  the  discovery  when  he  noticed  an  object  (appar- 
ently a  star)  where  none  had  previously  been  seen.  Piazzi 
called  the  new  planet,  which  was  of  small  dimensions,  Ceres. 

After  the  discovery  of  Ceres  had  been  made,  but  before 
the  news  of  it  had  reached  Germany  by  the  slow  processes 
of  communication  of  those  days,  the  philosopher  Hegel 
pubhshed  a  paper  in  which  he  clainaed  to  have  proved  by 


258    AN   INTRODUCTION   TO  ASTRONOMY  [ch.  viii,  160 

the  most  certain  and  conclusive  philosophical  reasoning  that 
there  were  no  new  planets,  and  he  ridiculed  his  astronomical 
colleagues  for  their  folly  in  searching  for  them. 

Piazzi  observed  Ceres  for  a  short  time  and  then  he  was 
taken  ill.  By  the  time  he  had  recovered,  the  earth  had 
moved  forward  in  its  orbit  to  a  position  from  which  the 
planetoid  could  no  longer  be  seen.  In  a  little  less  than  a 
year  the  earth  was  again  in  a  favorable  position  for  obser- 
vations of  Ceres,  but  the  problem  of  picking  it  up  out  of  the 
countless  stars  that  fill  the  sky,  and  from  which  it  could  not 
be  distinguished  except  by  its  motions,  was  almost  as  difficiilt 
as  that  of  making  the  original  discovery.  The  difficulty 
was  entirely  overcome  by  Gauss,  then  a  young  man  of  24, 
but  later  one  of  the  greatest  mathematicians  of  his  time,  for, 
under  the  stimulus  of  this  special  problem,  he  devised  a 
practical  method  of  detei^mining  the  elements  pf  the  orbit 
of  a  planet  from  only  three  observations.  After  the  ele- 
ments of  the  orbit  of  a  body  are  known,  its  position  can  be 
computed  at  any  time.  Gauss  determined  the  elements' of 
the  orbit  of  Ceres,  and  his  calculation  of  its  position  led  to  its 
rediscovery  on  the  last  day  of  the  year. 

On  March  28,  1802,  Olbers  discovered  a  second  planetoid, 
which  he  named  Pallas ;  on  September  2,  1804,  Harding 
found  Juno ;  and  on  March  29,  1807,  Olbers  picked  up  a 
fourth,  Vesta.  No  other  was  found  until  1845,  when  Hencke 
discovered  Astraea,  after  a  long  search  of  15  years.  In  1847 
three  more  were  discovered,  and  every  year  since  that  time 
at  least  one  has  been  discovered. 

In  1891  a  new  epoch  was  started  by  Wolf,  of  Heidelberg, 
who  discovered  a  planetoid  by  photography.  The  method 
is  simply  to  expose  a  plate  two  or  three  hours  with  the 
telescope  following  the  stars.  The  star  images  are  points, 
but  the  planetoids  leave  short  trails,  or  streaks.  Fig.  99, 
because  they  are  moving  among  the  stars.  There  are  now 
all  together  more  than  800  known  planetoids. 

After  the  first  two  planetoids  had  been  discovered  it  was 


CH.  viii,  160]  THE    SOLAR   SYSTEM    -  259 

supposed  that  they  might  be  simply  the  fragments  of  an 
original  large  planet  which  had  been  torn  to  pieces  by  an 
explosion.  If  such  were  the  case,  the  different  parts  in  their 
orbits  around  the  sun  would  all  pass  through  the  position 
occupied  by  the  planet  at  the  time  of  the  explosion ;  there- 
fore, for  some  time  the  search  for  new  planetoids  was  largely 
confined  to  the  regions  about  the  points  where  the  orbits 
of  Ceres  and  Pallas  intersect.     But  this  theory  of  their 


Fig.  99.  —  Photograph  of  stars  showing  a  planetoid  (Egeria)  trail  in  the 
center  of  the  plate.    Photographed  by  Parkhurst  at  the  Yerkes  Observatory. 

origin  has  been  completely  abandoned.  The  orbits  of  Eros 
and  two  other  planetoids  are  interior  to  the  orbit  of  Mars, 
while  many  are  within  75,000,000  miles  of  this  planet ;  on 
the  other  hand,  many  others  are  nearly  300,000,000  miles 
farther  out,  and  the  aphelia  of  four  are  even  beyond  the  orbit 
of  Jupiter.  Their  orbits  vary  in  shape  from  almost  perfect 
circles  to  elongated  ellipses  whose  eccentricities  are  0.35  to 
0.40.  The  average  eccentricity  of  their  orbits  is  about  0.14, 
or  approximately  twice  that  of  the  orbits  of  the  planets. 
Their  inclinations  to  the  ecliptic  range  all  the  way  from  zero 
to  35°,  with  an  average  of  about  9°. 


260   AN   INTRODUCTION   TO   ASTRONOMY    [ch.  viii,  160 

The  orbit's  of  the  planetoids  are  distributed  by  no  means 
uniformly  over  the  belt  which  they  occupy.  Kirkwood  long 
ago  called  attention  to  the  fact  that  the  planetoids  are  infre- 
quent, or  entirely  lacking,  at  the  distances  at  which  their 
periods  would  be  2!  h  h  ■  ■  •  of  Jupiter's  period.  The 
numerous  discoveries  since  the  application  of  photography 
have  still  further  emphasized  the  existence  of  these  remark- 
able gaps.  It  is  supposed  that  the  perturbations  by  Jupiter 
during  indefinite  ages  have  cleared  these  regions  of  the 
bodies  that  may  once  have  been  circulating  in  them,  but  the 
question  has  not  received  rigorous  mathematical  treatment. 

The  diameters  of  Ceres,  Pallas,  Vesta,  and  Juno  were 
measured  by  Barnard  with  the  36-inch  telescope  of  the  Lick 
Observatory,  and  he  found  that  they  are  respectively  485, 
304,  243,  and  118  miles.  There  are  probably  a  few  more 
whose  diameters  exceed  100  miles,  but  the  great  majority 
are  undoubtedly  much  smaller.  Probably  the  diameters  of 
the  faintest  of  those  which  have  been  photographed  do  not 
exceed  5  miles. 

By  1898  the  known  planetoids  were  so  numerous  and  their 
orbits  caused  so  much  trouble,  because  of  their  large  per- 
turbations by  Jupiter,  that  astronomers  were  on  the  point 
of  neglecting  them,  when  Witt,  of  Berlin,  found  one  within 
the  orbit  of  Mars,  which  he  named  Eros.  At  once  great 
interest  was  aroused.  On  examining  photographs  which 
had  been  taken  at  the  Harvard  College  Observatory  in 
1893,  1894,  and  1896,  the  image  of  Eros  was  found  several 
times,  and  from  these  positions  a  very  accurate  orbit  was 
computed  by  Chandler.  The  mean  distance  of  Eros  from 
the  sun  is  135,500,000  miles,  but  its  distance  varies  consid- 
erably because  its  orbit  has  the  high  eccentricity  of  0.22; 
its  inchnation  to  the  echptic  is  about  11°.  At  its  nearest, 
Eros  is  about  13,500,000  miles  from  the  earth,  and  then  con- 
ditions are  particularly  favorable  for  getting  the  scale  of  the 
solar  system  (Art.  156) ;  and  at  its  aphelion  it  is  24,000,000 
miles  beyond  the  orbit  of  Mars  (Fig.  96). 


CH.  VIII,  1,1]       ,       THE    SOLAR   SYSTEM  261 

Not  only  is  Eros  remarkable  because  of  the  position  of 
its  orbit,  but  in  February  and  March  of  1901  it  varied  in 
brightness  both  extensively  and  rapidly.  The  period  was 
2  hr.  38  min.,  and  at  minimum  its  ligl^t  was  less  than  one 
third  that  at  maximum.  By  May  the  variability  ceased. 
Several  suggestions  were  made  for  explaining  this  remarkable 
phenomenon,  such  as  that  the  planetoid  is  very  different  in 
reflecting  power  on  different  parts,  or  that  it  is  really  com- 
posed of  two  bodies  very  near  together,  revolving  so  that  the 
plane  of  their  orbit  at  certain  times  passes  through  the 
earth,  Ijut  the  cause  of  this  remarkable  variation  in  bright- 
ness is  as  yet  uncertain. 

161.  The  Question  of  Undiscovered  Planets.  —  The 
great  planets  Uranus  and  Neptune  have  been  discovered  in 
modern  times,  and  the  question  arises  if  there  may  not  be 
others  at  present  unknown.  Obviously  any  unknown  planets 
must  be  either  very  small,  or  very  near  the  sun,  or  beyond 
the  orbit  of  Neptune,  for  otherwise  they  already  would  have 
been  seen. 

The  perihelion  of  the  orbit  of  Mercury  moves  somewhat 
faster  than  it  would  if  this  planet  were  acted  on  only  by 
known  forces.  One  explanation  offered  for  this  pecuharity 
of  its  motion  is  that  it  may  be  disturbed  by  the  attraction  of 
a  planet  whose  orbit  Ues  between  it  and  the  suii.  A  planet 
in  this  position  would  be  observed  only  with  difficulty  be- 
cause its  elongation  from  the  sun  would  always  be  small. 
Half  a  century  ago  there  was  considerable  belief  in  the 
existence  of  an  intra-Mercurian  planet,  and  several  times 
it  was  supposed  one  had  been  observed.  But  photographs 
have  been  taken  of  the  region  around  the  sun  at  all  recent 
total  ecHpses,  and  in  no  case  has  any  object  within  the  orbit 
of  Mercury  been  found.  It  is  reasonably  certain  that  there 
is  no  object  in  this  region  more  than  20  miles  in  diameter. 

The  question  of  the  existence  of  trans-Neptunian  planets 
is  even  more  interesting  and  much  more  difficult  to  answer. 
Tljere  is  no  reason  to  suppose  that  Neptune  is  the  most  re- 


262    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  vjii,  161 

mote  planet,  and  the  gravitative  control  of  the  sun  extends 
enormously  beyond  it.  There  are  two  lines  of  evidence, 
besides  direct  observations,  that  bear  on  the  question.  If 
there  is  a  planet  of  considerable  mass  beyond  the  orbit  of 
Neptune,  it  will  in  time  make  its  presence  felt  by  its  pertur- 
bations of  Neptune.  Since  Neptune  was  discovered  it  has 
made  less  than  half  a  revolution,  and  the  fact  that  its  observed 
motion  so  far  agrees  with  theory  is  not  conclusive  evidence 
against  the  existence  of  a  planet  beyond.  In  fact,  there  are 
some  very  slight  residual  errors  in  the  theory  of  the  motion 
of  Uranus,  and  from  them  Todd  inferred  that  there  is 
probably  a  planet  revolving  at  the  distance  of  about  50  as- 
tronomical units  in  a  period  of  about  350  years.  The  con- 
clusion is  uncertain,  though  it  may  be  correct.  A  much 
more  elaborate  investigation  has  been  made  by  Lowell,  who 
finds  that  the  slight  discrepancies  in  the  motion  of  Uranus 
are  notably  reduced  by  the  assumption  of  the  existence  of 
a  planet  at  the  distance  of  44  astronomical  units  (period  290 
years)  whose  mass  is  greater  than  that  o'J  the  earth  and  less 
than  that  of  Neptune. 

It  will  be  seen  (Art.  196)  that  planets  sometimes  capture 
comets  and  reduce  their  orbits  so  that  their  aphelia  are 
near  the  orbits  of  their  captors.  Jupiter  has  a  large  family 
of  comets,  and  Saturn  and  Uranus  have  smaller  ones.  As 
far  back  as  1880,  Forbes,  of  Edinburgh,  inferred  from  a 
study  of  the  orbits  of  those  comets  whose  apheUa  are  beyond 
the  orbit  of  Neptune  that  there  are  two  remote  members  of 
the  solar  family  revolving  at  the  distances  of  100  and  300 
astronomical  units  in  the  immense  periods  of  1000  and  5000 
years.  W.  H.  Pickering  has  made  an  extensive  statistical 
study  of  the  orbits  of  comets  and  infers  the  probable  exist- 
ence of  three  or  four  trans-Neptunian  planets.  The  data 
are  so  uncertain  that  the  correctness  of  the  conclusion  is 
much  in  doubt. 

162.  The  Zodiacal  Light  and  the  Gegenschein.  —  The 
zodiacal  light  is  a  soft,  hazy  wedge  of  light  stretching  up 


CH.  viii,  162]  THE    SOLAR   SYSTEM  263 

from  the  horizon  along  the  ecliptic  just  as  twilight  is  ending 
or  as  dawn  is  beginning.  Its  base  is  20°  or  30°  wide  and  it 
,  generally  can  be  followed  90°  from  the  sun,  and  sometimes 
it  can  be  seen  as  a  narrow,  very  faint  band  3°  or  4°  wide  en- 
tirely aroimd  the  sky.  It  is  very  difficult  to  decide  precisely 
what  its  limits  are,  for  it  shades  very  gradually  from  an 
illumination  perhaps  a  httle  brighter  than  the  Milky  Way 
into  the  dark  sky. 

The  best  time  to  observe  the  zodiacal  hght  is  when  the 
echptic  is  nearly  perpendicular  tc  the  horizon,  for  then  it  is 
less  interfered  with  by  the  dense  lower  air.  In  the  spring 
the  sun  is  very  near  the  vernal  equinox.  At  this  time  of 
the  year  the  echptic  comes  up  after  sunset  from  the  western 
horizon  north  of  the  equator,  and  makes  a  large  angle  with 
the  horizon.  Consequently,  the  spring  months  are  most 
favorable  for  observing  the  zodiacal  light  in  the  evening,  and 
for  analogous  reasons  the  autumn  months  are  most  favorable 
for  observing  it  in  the  morning.  It  cannot  be  seen  in  strong 
moonhght. 

The  gegenschein,  or  counterglow,  is  a  very  faint  patch  of 
hght  in  the  sky  on  the  ecliptic  exactly  opposite  to  the  sim. 
It  is  oval  in  shape,  from  10°  to  20°  in  length  along  the  ecliptic, 
and  about  half  as  wide.  It  was  first  discovered  by  Brorsen 
in  1854,  and  later  it  was  found  independently  by  Backhouse 
and  Barnard.  It  is  so  excessively  faint  that  it  has  been 
observed  by  only  a  few  people. 

The  cause  of  the  gegenschein  is  not  certainly  known. 
It  has  been  suggested  that  it  is  a  sort  of  swelling  in  the 
zodiacal  band  which  appears  to  be  a  continuation  of  the 
zodiacal  light.  This  explanation  calls  for  an  explanation  of 
the  zodiacal  light,  which,  of  course,  might  well  be  independ- 
ently asked  for.  The  zodiacal  hght  is  almost  certainly  due 
to  the  reflection  of  hght  from  a  great  number  of  small  parti- 
cles circulating  around  the  sun  in  the  plane  of  the  earth's 
orbit,  and  extending  a  httle  beyond  the  orbit  of  the  earth. 
An  observer  at  0,  Fig.  100,  would  see  a  considerable  num- 


264    AN   INTRODUCTION    TO   ASTRONOMY   [ch.  viii,  162 


ber  of  these  illuminated  particles  above  his  horizon  H ;  and 

with  the  conditions  as  represented  in  the  diagram,  the  zodi- 

- .     acal  band  would   extend  faintly 

,  -    -  :  -'  -  -  ■     beyond  the  zenith  and  across  the 

sky. 

It  is  not  clear  from  this  theory 
of  the  zodiacal  Hght  why  there 
should  be  a  condensation  exactly 
opposite  the  sun.  But  at  a  point 
930,000  miles  from  the  earth, 
which  is  beyond  the  apex  of  its 
shadow,  there  is  a  region  where, 
in  consequence  of  the  combined 
forces  of  the  earth  and  sun, 
wandering  particles  tend  to  circulate  in  a  sort  of  dynamic 
whirlpool.  It  has  been  suggested  that  the  circulating  parti- 
cles which  produce  the  zodiacal  light  are  caught  in  this 
whirl  and  are  virtually  condensed  enough  to  produce  the 
observed  phenomenon  of  the  gegenschein. 


Fig.   100.  —  Explanation  of 
the  zodiacal  light. 


XII.  QUESTIONS 

1.  Which  of  the  methods  of  measuring  the  distance  from  the  earth 
to  the  sun  depend  upon  our  knowledge  of  the  size  of  the  earth,  and 
which  are  independent  of  it  ? 

2.  Make  a  single  drawing  showing  the  orbits  of  all  the  planets 
to  the  same  scale.  On  this  scale,  what  are  the  diameters  of  the  earth 
and  of  the  moon's  orbit  ? 

3.  If  the  sun  is  represented  by  a  globe  1  foot  in  diameter,  what 
would  be  the  diameters  and  distances  of  the  planets  on  the  same 
scale  ? 

4.  How  long  would  it  take  to  travel  a  distance  equal  to  that  from 
the  sun  to  the  earth  at  the  rate  of  60  miles  per  hour  ?  How  much 
would  it  cost  at  2  cents  per  mile  ? 

5.  The  magnitude  of  the  sun  as  seen  from  the  earth  is  —  26.7. 
What  is  its  magnitude  as  seen  from  Neptune  ?  As  seen  from  Nep- 
tune, how  many  times  brighter  is  the  sun  than  Sirius  ? 

6.  If  Jupiter  were  twice  as  far  from  the  sun,  how  much  fainter 
would  it  be  when  it  is  in  opposition  ? 


CH.  VIII,  162]  THE    SOLAR   SYSTEM  265 

7.  How  great  are  the  variations  in  the  distances  of  the  planets 
from  the  sun  which  are  due  to  the  eccentricities  of  their  orbits  ? 

8.  Suppose  the  earth  and  Neptune  were  in  a  line  between  the 
sun  and  the  nearest  star ,;  how  much  brighter  would  the  star  appear 
from  Neptune  than  from  the  earth  ? 

9.  In  what  respects  are  all  the  planets  similar  ?  In  what  respects 
are  the  four  inner  planets  similar  and  different  from  the  four  outer 
planets  ?  In  what  respects  are  the  four  outer  planets  similar,  and 
different  from  the  four  inner  planets  ? 

10.  Find  the  velocities  with  which  the  planets  move,  assuming 
their  orbits  are  circles. 

11-.  Find  the  next  dates  at  which  Mercury  and  Venus  will  have 
their  greatest  eastern  elongations,  and  at  which  Mars,  Jupiter,  and 
Saturn  will  be  in  opposition. 

12.  If  possible,  observe  the  zodiacal  Ught  and  describe  its  location 
and  characteristics. 


CHAPTER  IX 
THE   PLANETS 

I.   Mercury  and  Venus 

163.  The  Phases  of  Mercury  and  Venus.  —  The  inferior 
planets  Mercury  and  Venus  are  alike  in  several  respects  and 
may  conveniently  be  treated  together.  They  both  have 
phases  somewhat  analogous  to  those  of  the  moon.  When 
they  are  in  inferior  conjunction,  that  is,  at  A,  Fig.  101,  their 

dark   side   is   toward 

..-  ''        ^"-,-. ,  the    earth   and    their 

'■.'""---.  phase  is  new.     Since 

""-■-.,  the    orbits    of    these 

ce-— <^ <?^ ■■->t   planets    are    inclined 

SUN  I 

somewhat  to  the  plane 

of   the  ecliptic,  they 

-V  ,    '  do  not  in  general  pass 

^     ,„,      „^         ,      .  ,   .      ,     ,       across  the  sun's  disk. 

Fig.   101.  —  Phases  of  an  inferior  planet. 

If  they  do  not  make 
a  transit,  they  present  an  extremely  thin  crescent  when  they 
have  the  same  longitude  as  the  sun.  As  they  move  out 
from  A  toward  B. their  crescents  increase,  and  their  disks, 
as  seen  from  the  earth,  are  half  illuminated  when  they  have 
their  greatest  elongation  at  B.  During  their  motion  from 
inferior  conjunction  at  A  to  their  greatest  elongation  at  B, 
and  on  to  their  superior  conjunction  at  C,  their  distances 
from  the  earth  constantly  increase,  and  this  increase  of 
distance  to  a  considerable  extent  offsets  the  advantage 
arising  from  the  fact  that  a  larger  part  of  their  illuminated 
areas  are  visible.  In  order  that  an  inferior  planet  may  be 
seen,  not  only  must  its  illuminated  side  be  at  least  partly 

266 


CH.  IX,  163]  THE   PLANETS  '    267 

toward  the  earth,  but  it  must  not  be  too  nearly  in  a  line  with 
the  sun.  For  example,  a  planet  at  C,  Fig.  101,  has  its  il- 
luminated side  toward  the  earth,  but  it  is  invisible  because 
it  is  almost  exactly  in  the  same  direction  as  the  sun. 

The  variations  in  the  apparent  dimensions  of  Venus  are 
greater  than  those  of  Mercury  because,  when  Venus  is  near- 
est the  earth,  it  is  much  nearer  than  the  closest  approach  of 
Mercury,  and  when,  it  is  farthest  from  the  earth,  it  is  much 
farther  than  the  most  remote  point  in  Mercury's  orbit. 
At  the  time  of  inferior  conjunction  the  distance  of  Venus  is 
25,700,000  miles,  while  that  of  Mercury  is  56,900,000  miles ; 
and  at  superior  conjunction  their  respective  distances  are 
160,100,000  and  128,900,000  miles.  These  numbers  are 
modified  somewhat  by  the  eccentricities  of  the  orbits  of 
these  three  bodies,  and  especially  by  the  large  eccentricity 
of  the  orbit  of  Mercury. 

Mercury  and  Venus  transit  across  the  sun's  disk  only 
when  they  pass  through  inferior  conjunction  with  the  sun 
near  one  of  the  nodes  of  their  orbits.  The  sun  is  near  the 
nodes  of  Mercury's  orbit  in  May  and  November,  and  con- 
sequently this  planet  transits  the  sun  only  if  it  is  in  inferior 
conjunction  at  one  of  these  times.  Since  there  is  no  simple 
relation  between  the  period  of  Mercury  and  that  of  the 
earth,  the  transits  of  Mercury  do  not  occur  very  frequently. 
A  transit  of  Mercury  is  followed  by  another  at  the  same  node 
of  its  orbit  after  an  interval  of  7,  13,  or  46  years,  according 
to  circumstances,  for  these  periods  are  respectively  very 
nearly  22,  41,  and  145  synodical  revolutions  of  the  planet. 
Moreover,  there  may  be  transits  also  when  Mercury  is  near 
the  other  node  of  its  orbit.  The  next  transits  of  Mercury  will 
occur  on  May  7,  1924,  and  on  November  8,  1927.  Mercury 
is  so  small  that  its  transits  can  be  observed  only  with  a 
telescope. 

The  transits  of  Venus,  which  occur  in  June  and  December, 
are  even  more  infrequent  than  those  of  Mercury.  The 
transits  of  Venus  occur  in  cycles  whose  intervals  are,  starting 


268     AN   rNTRODUCTION   TO   ASTRONOMY     [ch.  ix,  163 

with  a  June  transit,  8, 105.5, 8,  and  121.5  years.  The  last  two 
transits  of  Venus  occurred  on  December  8,  1874,  and  on 
Decerriber  6,  1882.  The  next  two  will  occur  on  June  8, 
2004,  and  on  June  5,  2012. 

'  The  chief  scientific  uses  of  the  transits  of  Mercury  and 
Venus  are  that  they  give  a  means  of  determining  the  posi- 
tions of  these  planets,  they  make  it  possible  to  investigate 
their  atmospheres,  and  they  were  formerly  used  indirectly 
for  determining  the  scale  of  the  solar  system  (Art.  156). 

164.  The  Albedoes  and  Atmospheres  of  Mercury  and 
Venus.  —  The  albedo  of  a  body  is  the  ratio  of  the  light  which 
it  reflects  to  that  which  it  receives.  The  amount  of  Ught 
reflected  depends' to  a  considerable  extent  upon  whether  or 
not  the  body  is  surrounded  by  a  cloud-filled  atmosphere.  A 
body  which  has  no  atmosphere  and  a  rough  and  broken 
surface,  like  the  moon,  has  a  low  albedo,  while  one  covered 
with  an  atmosphere,  especially  if  it  is  filled  with  partially 
condensed  water  vapor,  has  a  higher  refiecting  power.  Every 
one  is  familiar  with  the  fact  that  the  thunderheads  which 
often  appear  in  the  summer  sky  shine  as  white  as  snow 
when  illuminated  fully  by  the  sun's  rays.  It  was  found  by 
Abbott  that  their  albedo  is  about  0.65.  If  an  observer  could 
see  the  earth  from  the  outside,  its  brightest  parts  would 
undoubtedly  be  those  portions  of  its  surface  which  are 
covered  either  by  clouds  or  by  snow. 

The  albedo  of  Mercury,  according  to  the  careful  work  of 
Miiller,  of  Potsdam,  is  about  0.07,  while  that  of  Venus  is 
0.60.  This  is  presumptive  evidence  that  the  atmosphere 
of  Mercury  is  either  very  thin  or  entirely  absent,  and  that 
that  of  Venus  is  abundant. 

It  follows  from  the  kinetic  theory  of  gases  (Art.  32)  and 
the  low  surface  gravity  of  Mercury  (Art.  158,  Table  V)  that 
Mercury  probably  does  not  have  sufficient  gravitative  con- 
trol to  retain  a  very  extensive  atmospheric  envelope.  This 
inference  is  confirmed  by  the  fact  that,  when  Mercury 
transits  the  sun,  no  bright  ring  is  seen  around  it  such  as  would 


CH.  IX,  165]  THE  ■  PLANETS  269 

be  observed  if  it  were  surrounded  by  any  considerable  atmos- 
phere. Moreover,  Miiller  found  that  the  amount  of  light 
received  from  Mercury  at  its  various  phases  proves  that  it 
is  reflected  from  a  solid,  uneven  surface.  Therefore  there  is 
abundant  justification  for  the  concliision  that  Mercury  has 
an  extremely  tenuous  atmosphere,  or  perhaps  none  at  all. 

The  evidence  regarding  the  atmosphere  of  Venus  is  just 
the  opposite  of  that  encountered  in  the  case  of  Mercury. 
Its  considerable  mass  and  surface  gravity,  approximating 
those  of  the  earth,  naturally  lead  to  the  conclusion  that  it 
can  retain  an  atmosphere  comparable  to  our  own.  But  the 
conclusions  do  not  rest  alone  upon  such  general  arguments ; 
for,  when  Venus  transits  the  sun,  its  disk  is  seen  to  be  sur- 
roimded  by  an  illuminated  atmospheric  ring.  Besides  this, 
when  it  is  not  in  transit,  but  near  inferior  conjunction,  the 
illuminated  ring  of  its  atmosphere  is  sometimes  seen  to 
extend  considerably  beyond  the  horns  of  the  crescent.  Also, 
the  briUiancy  of  Venus  decreases  somewhat  from  the  center 
toward  the  margin  of  its  disk  where  the  absorption  of  light 
would  naturally  be  '  the  greatest.  Spectroscopic  observa- 
tions, which  are  as  yet  somewhat  doubtful,  point  to  the 
conclusion  that  the  atmosphere  of  Venus  contains  water 
vapor.  Taking  all  the  evidence  together,  we  are  justified  in 
the  conclusion  that  Venus  has  an  atmospheric  .envelope 
corresponding  in  extent,  and  possibly  in  composition,  to  that 
of  the  earth. 

165.  The  Surface  Markings  and  Rotation  of  Mercury.  — 
The  first  astronomer  to  observe  systematically  and  continu- 
ously the  surface  markings  of  the  sun,  moon,  and  planets 
was  Sxihroter  (1745-1816).  He  was  an  astronomer  of  rare 
enthusiasm  dnd  great  patience,  but  seems  sometimes  to 
have  been  led  by  his  lively  imagination  to  erroneous 
conclusions. 

Schroter  concluded  from  observations  of  Mercury  made 
in  1800,  that  the  period  of  rotation  of  this  planet  is  24  hours 
and  4  minutes.    This  result  was  quite  generally  accepj.ed 


270     AN   INTRODUCTION   TO   ASTRONOMY     (ch.  ix,  165 

until  after  ScHiaparelli  took  up  his  systematic  observations 
of  the  planets,  at  Milan,  about  1880.  SchiaparelU  found 
that  Mercury  could  J)e  much  better  seen  in  the  daytime, 
when  it  was  near  the  meridian,  than  in  the  evening,  because 
the  illumination  of  the  sky  was  found  to  be  a  much  less 
serious  obstacle  than  the  absorption  and  irregularities  of 
refraction  which  were  encountered  when  Mercury  was  near 
the  horizon.  His  experience  in  this  matter  has  been  con- 
firmed by  later  astronomers. 

Schiaparelli  came  to  the  conclusion,  from  elusive  and 
vague  markings  on  the  planet,  that  its  axis  is  essentially 
perpendicular  to  the  plane  of  its  orbit,  and  that  its  periods 
of  rotation  and  revolution  are  the  same.  These  results  are 
agreed  to  by  Lowell,  who  has  carefully  observed  the  planet 
with  an  excellent  24-inch  telescope  at  Flagstaff,  Ariz. 
Although 'the  observations  are  very  difficult,  we  are  perhaps 
entitled  to  conclude  that  the  same  face  of  Mercury  is  always 
toward  the  sun. 

166.  The  Seasons  of  Mercury.  —  If  the  period  of  rota- 
tion of  Mercury  is  the  same  as  that  of  its  revolution,  its  sea- 
sons are  due  entirely  to  its  varying  distance  from  the  sun 
and  the  varying  rates  at  which  it  moves  in  its  orbit  in  har- 
mony with  the  law  of  areas.  The  eccentricity  of  the  orbit 
of  Mercury  is  so  great  that  at  perihelion  its  distance  from  the 
sun  is  only  two  thirds  of  that  at  aphelion.  Since  the  amount 
of  light  and  heat  the  planet  receives  varies  inversely  as 
the  square  of  its  distance  from  the  sim,  it  follows  that  the 
illumination  of  Mercury  at  aphelion  is  only  four  ninths  of 
that  at  perihelion.  It  is  obvious  that  this  fatcior-alone  would 
make  an  important  seasonal  change. 

Whatever  the  period  of  rotation  of  Mercury  may  be,  its 
rate  of  rotation  must  be  essentially  uniform.  Since  it 
moves  in  its  orbit  so  as  to  fulfill  the  law  of  areas,  its  motion 
of  revolution  is  sometimes  faster  and  sometime^  slower  than 
the  average.  The  result  of  this  is  that  not  exactly  the  same 
side  of  Mercury  is  always  toward  the  sun,  even  if  its  periods 


CH.  IX,  167]  THE   PLANETS  271 

of  revolution  and  rotation  are  the  same.  The  mathematical 
discussion  shows  that,  at  its  greatest,  it  is  23°.7  ahead  of  its 
mean  position  in  its  orbit,  and  consequently,  at  such  a  time, 
the  s\m  shines  around  the  surface  of  Mercury  23°.7  beyond 
the  point  its  rays  would  reach  if  its  orbit  were  strictly  a 
circle.  Similarly,  the  planet  at  times  gets  23°.7  behind  its 
mean  position.  That  is.  Mercury  has  a  Ubration  (Art.  129) 
of  23°.7.  If  Mercury's  period  of  rotation  equals  its  period 
of  revolution,  there  are,  therefore,  132°.6  of  longitude  on 
the  planet  on  which  the  sun  always  shines,  an  equal  amount 
on  which  it  never  shines,  and  two  zones  47°.4  wide  in  which 
there  is  alternating  day  and  night  with  a  period  equal  to 
the  period  of  the  planet's  revolution  around  the  sun. 

If  the  periods  of  rotation  and  revolution  of  Mercury  are 
the  same,  the  side  toward  the  sun  is  perpetually  subject  to 
its  burning  rays,  which  are  approximately  ten  times  as 
intense  as  they  are  at  the  distance  of  the  earth,  and,  more- 
over, they  are  never  cut  off  by  clouds  or  reduced  by  an 
appreciable  atmosphere.  The  only  possible  conclusion  is 
that  the  temperature  of  this  portion  of  the  planet's  surface 
is  very  high.  On  the  sjde  on  which  the  sun  never  shines  the 
temperature  must  be  extremely  low,  for  there  is  no  atmos- 
phere to  carry  heat  to  it  from  the  warm  side  or  to  hold  in 
that  which  may  be  conducted  to  the  surface  from  the  interior 
of  the  planet.  The  intermediate  zones  are  subject  to  alter- 
nations of  heat  and  cold  with  a  period  equal  to  the  period 
of  revolution  of  the  planet,  and  every  temperature  between 
the  two  extremes  is  found  in  some  zone. 

167.  The  Surface  Markings  and'  Rotation  of  Venus.  — 
The  history  of  the  observations  of  Venus  and  the  conclu- 
sions regarding  its  rotation  are  almost  the  same  as  in  the 
case  of  IVtercury.  As  early  as  1740  J.  J.  Cassini  inferred  from 
the  observations  of  his  predecessors  that  Venus  rotates  on  its 
axis  in  23  hours  and  20  minutes.  About  1790  Schroter  con- 
cluded that  its  rotation  period  is  about  23  hours  and  21 
minutes,  and  that  the  inclination  of  the  plane  of  its  equator 


272     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ix,  167 

to  that  of  its  orbit  is  53°.  These  results  were  generally 
accepted  until  1880,  when  SchiapareUi  announced  that  Venus, 
like  Mercury,  always  has  the  same  face  toward  the  sun. 

The  observations  of  SchiapareUi  were  verified  by  himself 
in  1895,  and  they  have  been  more  or  less  definitely  confirmed 
by  Perrotin,  Tacchini,  Mascari,  Cerulli,  Lowell,  and  others. 
However,  it  must  be  remarked  that  the  atmosphere  interferes 
with  seeing  the  surface  of  Venus  and  that  the  observations 
are  very  doubtful.  Moreover,  recent  direct  observations 
by  a  number  of  experienced  astronomers  point  to  a  period  of 
rotation  of  about  23  or  24  hours. 

The  spectroscope  can  also  be  applied  under  favorable 
conditions  to  determine  the  rate  at  which  a  body  rotates. 
In  1900  Belopolsky  concluded  from  observations  of  this  sort 
that  the  period  of  rotation  of  Venus  is  short.  More  accurate 
observations  by  Shpher,  at  the  Lowell  Observatory,  show  no 
evidence  of  a  short  period  of  rotation.  The  preponderance 
of  evidence  seems  to  be  in  favor  of  the  long  period  of  rotation, 
but  the  conclusion  is  at  present  very  uncertain. 

168.  The  Seasons  of  Venus.  —  The  character  of  the  sea- 
sons of  Venus  depends  very  much  upon  whether  the  planet's 
period  of  rotation  is  approximately  24  hours  or  is  equal 
to  its  period  of  revolution.  If  the  planet  rotates  in  the 
shorter  period  and  if  its  equator  is  somewhat  inclined  to 
the  plane  of  its  orbit,  the  seasons  must  be  quite  similar  to 
those  of  the  earth,  though  the  temperature  is  probably  some- 
what higher  because  the  planet  is  nearer  to  the  sim.  On  the 
other  hand,  if  the  same  face  of  Venus  is  always  toward  the 
sun,  the  conditions  must  be  more  hke  those  on  Mercury, 
though  the  range  of  temperatures  cannot  be  so  extreme 
because  its  atmosphere  reduces  the  temperature  on  the  side 
toward  the  sun  and  raises  it  on  the  opposite  side  by  carrying 
heat  from  the  warmer  side  to  the  cooler. 

Suppose  the  periods  of  rotation  and  revolution  of  Venus  are 
equal.  Since  the  orbit  of  Venus  is  very  nearly  circular,  it  is 
subject  to  only  a  small  libration  and  only  a  very  narrow  zone 


CH.  IX,  169]  THE   PLANETS  273 

around  it  has  alternately  day  and  night.  The  position  of 
the  sun  in  its  sky  is  nearly  fixed  and  the  climate  at  every 
place  on  its  surface  is  remarkably  uniform.  There  must  be  a 
system  of  atmospheric  currents  of  a  regularity  not  known  on 
the  earth,  and  it  has  been  suggested  that  all  the  water  on 
the  planet  was  long  ago  carried  to  the  dark  side  in  clouds  and 
precipitated  there  as  snow.  This  conclusion  is  not  neces- 
sarily true,  for  it  seems  likely  that  the  air  would  ascend  on 
the  heated  side  and  lose  its  moisture  by  precipitation  before 
the  high  currents  which  would  go  to  the  dark  side  had  pro- 
ceeded far  on  their  way. 

Considered  as  a  whole,  Venus  is  more  Uke  the  earth  than 
any  other  planet ;  and,  so  far  as  can  be  determined,  it  is  in 
a  condition  in  which  life  can  flourish.  In  fact,  if  any  other 
planet  than  the  earth  is  inhabited,  that  one  is  probably 
Venus.  It  must  be  added,  however,  that  there  is  no  direct 
evidence  whatever  to  support  the  supposition  that  there 
is  life  upon  its  surface. 

II.  Mass 

169.  The  Satellites  of  Mars.  —  In  August,  1877,  Asaph 
Hall,  at  Washington,  discovered  two  very  small  sateUites 
revolving  eastward  around  Mars,  sensibly  in  the  plane  of  its 
equator.  They  are  so  minute  and  so  near  the  bright  planet 
that  they  can  be  seen  only  with  a  large  telescope,  and  usually 
it  is  advantageous,  when  observing  them,  to  obscure  Mars 
by  a  small  screen  placed  in  the  fodal  plane.  These  satellites 
are  called  Phobos  and  Deimos.  The  only  way  of  determin- 
ing their  dimensions  is  from  the  amount  of  fight  they  reflect 
to  the  earth.  Though  Phobos  is  considerably  brighter 
than  Deimos,  its  diameter  probably  does  not  exceed  10  miles. 

Not  only  are  the  satellites  of  Mars  very  small,  but  in  other 
respects  they  present  only  a  rough  analogy  to  the  moon 
revolving  around  the  earth.  The  distance  of  Phobos  from 
the  center  of  Mars  is  only  5850  miles,  while  that  of  Deimos 
is  14,650  miles.     That  is,  Phobos  is  only  3680  miles  from  the 


274     AN   INTRODUCTION   TO  ASTRONOMY     [ch.  ix,  169 

surface  of  the  planet.  The  curvature  of  the  planet's  surface 
is  such  that  Phobos  could  not  be  seen  by  an  observer  from 
latitudes  greater  than  68°  15'  north  or  south  of  the  planet's 
equator.  The  relative  dimensions  of  Mars  and  the  orbits 
of  its  satellites  are  shown  in  Fig.  102. 

As  was  seen  in  Art.  154,  the  period  of  a  sateUite  depends 
upon  the  mass  of  the  planet  around  which  it  revolves  and 

upon  its   distance  from 

opeff_oFDo«ay^  the  planet's  center.   Not- 

,'''  ~"-,^  withstanding  the    small 

mass  of  Mars,  its  satel- 

/  noFp^  \        htes   are  so    close  that 

/  >'       ~-^  \      their  periods  of  revolu- 

©',      tion  are  very  short,  the 
;  I     period  of  Phobos  being 

;      7  hrs.    39  m.   and  that 
\  ^-  ....-''  1      of  Deimos  being  30  hrs. 

18  m.      Since  Mars  ro- 
tates on  its  axis  in  24  hrs. 
"^'•.  ,-•-''  and  37  m.,  Phobos  makes. 

more  than  3  revolutions 

Fig.    102.  —  Mars  and  the  orbits  of  its         ,  . ,       ,  ,  .  ,  . 

sateUites.  while    Mars    is    making 

one  rotation.  It  there- 
fore rises  in  the  west,  passes  eastward  across  the  sky,  and 
sets  in  the  east.  Here  is  an  example  in  which  the  direction 
of  apparent  motion  and  actual  motion  are  the  same.  The 
period  of  Phobos  from  meridian  to  meridian  is  11  hrs.  and 
7  m.  On  the  other  hand,  Deimos  rises  in  the  east  and  sets 
in  the  west  with  a  period  from  meridian  to  meridian  of 
131  hrs.  and  14  m. 

170.  The  Rotation  of  Mars.  —  In  1666  Hooke,  an  English 
observer,  and  Cassini,  at  Paris,  saw  dark  streaks  on  the 
ruddy  disk  of  Mars,  and  these  features  of  the  planet's  sur- 
face are  so  definite  and  permanent  that  even  to-day  astrono- 
mers can  recognize  the  objects  which  these  men  observed 
and  drew.     Some  of  them  are  shown  in  Fig.  103,  which  is  a 


CH.  IX,  170]  THE   PLANETS  275 

series  of  9  photographs,  taken  one  after  the  other  at  short 
intervals,  by  Barnard,  at  the  Yerkes  Observatory.  By 
comparing  observations  at  one  time  with  those  made  at  a 
later  date  the  period  of  rotation  of  the  planet  can  be  found. 
In  fact,  considerable  rotation  is  observable  in  the  short 
interval  covered  by  the  photographs  iri  Fig.  103.     Hooke 


Fig.   10.3.  — Mars.    Photoffraphed  bv  Barnard  with  the  40-inch  telescope  of 
the  Yerkes  Observatory,  Sept.  28,  1909. 

and  Cassini  soon  discovered  that  Mars  turns  on  its  axis  in 
a  perioti  of  a  httle  more  than  24  hrs.  By  comparing  their 
observations  with  those  of  the  present  day  it  is  found  that 
its  period  of  rotation  is  24  hrs.  37  m.  22.7  sees.  The 
high  order  of  accuracy  of  this  result  is  a  consequence  of  the 
fact  that  the  importance  of  the  errors  of  the  observations 
diminishes  as  the  time  over  which  they  extend  increases. 

The  inclination  of  the  plane  of  the  equator  of  Mars  to 
the  plane  of  its  orbit, is  between  23°  and  24°.  The  inclina- 
tion cannot  be  determined  as  accurately  as  the  period  of 


276     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ix,  170 

rotation  because  the  only  advantage  of  a  long  series  of 
observations  consists  in  their  number.  But,  in  spite  of  its 
uncertainty,  the  obliquity  of  the  ecliptic  of  Mars  to  its 
equator  is  certainly  approximately  equal  to  that  of  the 
earth,  and,  consequently,  the  seasonal  changes  are  quali- 
tatively much  hke  those  of  the  earth.  One  important 
difference  is  that  the  period  of  Mars  is  about  23  months, 
and,  therefore,  while  its  day  is  only  a  little  longer  than  that 
of  the  earth,  its  year  is  nearly  twice  as  long.  It  is  not  meant 
to  imply  by  these  statements  that  the  chmate  of  Mars  is 
similar  to  that  of  the  earth.  Its  distance  from  the  sun  is 
so  much  greater  that  the  amount  of  light  and  heat  it  receives 
per  unit  area  is  only  about  0.43  of  that  which  the  earth 
receives. 

171.  The  Albedo  and  Atmosphere  of  Mars.  —  According 
to  the  observations  of  Miiller,  the  albedo  of  Mars  is  0.15, 
which  indicates  probably  a  thin  atmosphere  on  the  planet. 

The  surface  gravity  of  Mars  is  only  0.36  that  of  the  earth, 
and,  consequently,  it  would  be  expected  on  the  basis  of  the 
kinetic  theory  of  gases  that  it  might  retain  some  atmosphere, 
though  not  a  very  extensive  one.  Direct  observations  of 
the  planet  confirm  this  conclusion.  In  the  first  place,  its 
surface  can  nearly  always  be  seen  without  appreciable  inter- 
ference from  atmospheric  phenomena.  If  the  earth  were 
seen  from  a  distant  planet,  such  as  Venus,  not  only  would 
the  clouds  now  and  then  entirely  shut  off  its  surface  from 
view,  but  the  reflection  and  absorption  of  light  in  regions 
where  there  were  no  clouds  would  probably  make  it  impos- 
sible to  see  distinctly  anything  on  its  surface. 

The  fact  that  Mars  has  a  rare  atmosphere  is  also  proved 
by  the  suddenness  with  which  it  cuts  off  the  light  from 
stars  when  it  passes  between  them  and  the  earth.  Those 
planets  which  have  extensive  atmospheres,  such  as  Jupiter, 
extinguish  the  Hght  from  the  stars  more  gradually.  If  the 
atmosphere  of  Mars,  relatively  to  its  mass,  were  of  the  same 
density  as  that  of  the  earth,  it  would  be  rarer  at  the  surface 


CH.  IX,  172]  THE   PLANETS  277 

of  the  planet  than  our  atmosphere  is  at  the  top  of  the  loftiest 
mountains.  , 

A  number  of  Unes  of  evidence  have  been  given  for  the 
conclusion  that  the  atmosphere  of  Mars  is  not  extensive. 
The  question  occasionally  arises  whether  it  has  any  atmos- 
phere at  all.  The  answer  to  this  must  be  in  the  affirmative, 
because  faint  clouds,  possibly  of  dust  or  mist,  have  often 
been  observed  on  its  surface.  They  are  very  common  along 
the  borders  of  the  bright  caps  which  cover  its  poles.  Another 
related  phenomenon  which  is  very  remarkable  and  not 
easy  to  explain  is  that,  sometimes  for  considerable  periods, 
the  planet's  whole  disk  is  dim  and  obscure  as  though  covered 
by  a  thin  mist. 
While  the  cause 
of  this  obscura- 
tion is  not 
known,  it  i's  sup- 
posed that  it  is 
a  phenomenon 
of    the     atmos- 

"  Fig.  104.  —  Barnard's  drawings  of  Mars. 

planet.     Besides 

this.  Mars  undergoes  seasonal  changes,  not  only  in  the  polar 
caps,  which  will  be  considered  in  the  next  article,  but  also 
even  in  conspicuous  markings  of  other  types.  Figure  104 
gives  three  drawings  of  the  same  side  of  Mars  by  Barnard, 
on  September  23,  October  22,  and  October  28, 1894,  showing 
notable  temporary  changes  in  its  appearance. 

172.  The  Polar  Caps  and  the  Temperature  of  Mars.  — 
The  surface  of  Mars  on  the  whole  is  dull  brick-red  in  color, 
but  its  polar  regions  during  their  winter  seasons  are  covered 
with  snow-white  mantles.  One  of  these  so-called  polar 
caps  sometimes  develops  in  the  course  of  two  or  three  days 
over  an  area  reaching  down  from  the  pole  25°  to  35°;  it 
remains  undiminished  in  brilliancy  during  the  long  winter 
of  the  planet;    and,  as  the  spring  advances,   it   gradually 


278     AN   INTRODUCTION   TO   ASTRONOMY     [ch.  ix,  172 


diminishes  in  size,  contracting  first  around  the  edges;  it 
then  breaks  up  more  or  less,  and  it  sometimes  entirely  dis- 
appears in  the  late  summer. 

After  the  southern  polar  cap  has  shrunk  to  the  dimensions 
given  by  Barnard's  observation  of  August  13,  1894,  Fig. 

105,  an  elongated  white 
patch  is  found  to  be  left 
behind  the  retreating  white 
sheet.  The  same  thing  was 
observed  in  the  same  place 
at  the  corresponding  Mar- 
tian season  in  1892,  and  also 
at  later  oppositions.  This 
means  that  the  phenome- 
non is  not  an  accident,  but 
that  it  depends  upon  the 
nature  of  the  surface-  of 
Mars.  Barnard  has  sug- 
gested that  there  may  be 
an  elevated  region  in  tjie 
place  on  which  the  spot  is 
observed  where  the  snow 
or  frost  remains  until  after 
it  has  entirely  disappeared 
in  the  valleys.  At  any  rate, 
this  phenomenon  strongly 
points  to  the  conclusion 
that  there  are  considerable 
irregularities  in  the/Surface 
of  Mars,  though  on  the 
whole  it  is  probably  much 
smoother  than  the  earth. 
This  is  an  important  point 
which  must  be  borne  in 
mind  in  interpreting  other  things  observed  upon  the  surface 
of  the  planet. 


Fig.  105. 


—  Disappearance  of  polar  cap 
of  Mars  (Barnard). 


CH.  IX,  172]  THE   PLANETS  279 

The  polar  cap  around  the  south  pole  of  Mars  has  been 
more  thoroughly  studied  than  the  one  at  the  north  pole 
because  the  south  pole  is  tijrned  toward  the  earth  when  Mars 
is  in  opposition  near  the  perihelion  point  of  its  orbit.  The 
eccentricity  of  the  orbit  of  this  planet  is  so  great  that  its 
distance  from  the  orbit  of  the  earth  when  it  is  at  its  perihelion 
(which  is  near  the  aphelion  of  the  earth's  orbit)  is  more  than 
23,000,000  miles  less  than  when  it  is  at  its  aphelion.  How- 
ever, in  the  course  of  inunense  time  the  mutual  perturbations 
of  the  planets  will  so  change  the  orbit  of  Mars  that  its  north- 
ern polar  region  will  be  more  favorably  situated  for  observa- 
tions from  the  earth  than  its  southern. 

If  the  polar  caps  of  Mars  are  due  to  snow,  there  must  be 
water  vapor  in  its  atmosphere.  The  spectroscope  is  an 
instrument  which  under  suitable  conditions  enables  the 
astronomer  to  determine  the  constitution  of  the,  atmosphere 
of  a  celestial  body  from  which  he  receives  light.  Mars  is 
not  well  adapted  to  the  purpose  because,  in  the  first  place, 
the  light  received  from  it  is  only  reflected  sunUght  which 
may  have  traversed  more  or  less  of  its  shallow  and  tenuous 
atmosphere;  and,  in  the  second  place,  the  atmosphere  of 
the  earth  itself  contains  usually  a  large  amount  of  water 
vapor.  It  is  not  easy  to  make  sure  that  the  absorption 
spectral  Unes  (Art.  225)  may  not  be  due  altogether  to  the 
water  vapor  in  the  earth's  atmosphere. 

The  early  spectroscopic  investigations  of  Huggins  and 
Vogel  pointed  toward  the  existence  of  water  on  Mars ;  the 
later  ones  by  Keeler  and  Campbell,  with  much  more  powerful 
instruments  and  under  better  atmospheric  conditions,  gave 
the  opposite  result ;  but  the  spectograms  obtained  by  SUpher 
at  the  Lowell  Observatory,  under  exceptionally  favorable 
instrumental  and  cUmatic  conditions,  again  indicate  water 
on  Mars.  In  view  of  the  difficulties  of  the  problem,  a  nega- 
tive result  could  scarcely  be  regarded  as  being  conclusive 
evidence  of  the  entire  absence  of  water  on  Mars,  while 
evidence  of  a  small  amount  of  water  vapor  in  its  atmosphere 


280      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  ix,  172 

is  not  unreasonable  and  is  quite  in  harmony  with  the  phe- 
nomena of  its  polar  caps. 

The  distance  of  Mars  from  the  sun  is  so  great  that  it 
receives  only  about  0.43  as  much  hght  and  heat  per  unit 
area  as  is  received  by  the  earth.  The  question  then  arises 
how  its  polar  caps  can  nearly,  or  entirely,  disappear,  while 
the  poles  of  the  earth  are  perpetually  buried  in  ice  and 
snow.  The  responses  to  this  question  have  been  various, 
many  of  them  ignoring  the  fundamental  physical  principles 
on  which  a  correct  answer  must  be  based. 

In  the  first  place,  consider  the  problem  of  determining 
what  the  average  temperature  of  Mars  would  be  if  its  atmos- 
phere and  surface  structure  were  exactly  like  those  of  the 
earth.  That  is,  let  us  find  what  the  temperature  of  the  earth 
would  be  if  its  distance  from  the  sun  were  equal  to  that  of 
Mars.  The  amount  of  heat  a  planet  radiates  into  space  on 
the  average  equals  that  which  it  receives,  for  otherwise  its 
temperature  would  continually  increase  or  diminish.  There- 
fore, the  amount  of  heat  Mars  radiates  per  unit  area  is  on 
the  average  0.43  of  that  radiated  per  unit  area  by  the  earth. 
Now  the  amount  of  heat  a  body  radiates  depends  on  the 
character  of  its  surface  and  on  its  temperature.  In  this 
calculation  the  surfaces  of  Mars  and  the  earth  are  assumed 
to  be  alike.  According  to  Stefan's  law,  which  has  been  veri- 
fied both  theoretically  and  experimentally,  the  radiation  of 
a  black  body  varies  as  the  fourth  power  of  its  absolute 
temperature.  Or,  the  absolute  temperatures  of  two  black 
bodies  are  as  the  fourth  roots  of  their  rates  of  radiation. 

Now  apply  this  proportion  to  the  case  of  Mars  and  the 
earth.  On  the  Fahrenheit  scale  the  mean  annual  surface 
temperature  of  the  whole  earth  is  about  60°,  or  28°  above 
freezing.  The  absolute  zero  on  the  Fahrenheit  scale  is 
491°  below  freezing.  Therefore,  the  mean  temperature  of 
the  earth  on  the  Fahrenheit  scale  counted  from  the  absolute 
zero  is  about  491° -f  28°  =  519°.  Let  x  represent "  the 
absolute  temperature  of  Mars ;  then,  under  the  assumption 


CH.  IX,  172]  THE   PLANETS  281 

that  its  surface  is  like  that  of  the  earth,  the  proportion  becomes 

X  :  519  =  -v/OiiS  :  </I, 

from  which, it  is  found  that  x  =  420°.  That  is,  under  these 
hypotheses,  the  mean  surface  temperature  of  Mars  comes 
out  491°-420°  =  71°  below  freezing,  or  71°-32°  =  39° 
below  zero  Fahrenheit. 

The  results  just  obtained  can  lay  no  claim  to  any  par- 
ticular degree  of  accuracy  because  of  the  uncertain  hypotheses 
on  which  they  rest.  But  it  does  not  seem  that  the  hypothesis 
that  the  surfaces  of  Mars  and  the  earth  Badiate  similarly 
can  be  enough  in  error  to  change  the  results  by  very  many 
degrees.  If  the  atmosphere  of  Mars  were  of  the  same  con- 
stitution as  that  of  the  earth,  but  simply  more  tenuous,  its 
actual  temperature  would  be  lower  than  that  found  by  the 
computation.  On  the  other  hand,  if  the  atmosphere  of 
Mars  contained  an  abundance  of  gases  which  strongly 
absorb  and  retain  heat,  such  as  water  vapor  and  carbon 
dioxide,  its  mean  temperature  might  be  considerably  above 
—  39°.  But,  taking  all  these  possibilities  into  consideration, 
it  seems  reasonably  certain  that  the  mean  temperature  of 
Mars  is  considerably  below  zero  Fahrenheit.  The  question 
then  arises  how  its  polar  caps  can  almost,  or  entirely, 
disappear  each  summer. 

The  fundamental  principles  on  which  precipitation  and 
evaporation  depend  can  be  understood  by  considering  these 
phenomena  in  ordinary  meteorology.  If  there  is  a  large 
quantity  of  water  vapor  in  the  air  and  the  temperature 
falls,  there  is  precipitation  before  the  freezing  point  is  reached, 
and  the  result  is  rain.  On  the  other  hand,  if  the  amount  of 
water  vapor  in  the  air  is  small,  there  is  no  precipitation 
until  after  the  temperature  has  descended  below  the  freezing 
point  of  water.  In  this  case  when  precipitation  occurs  it 
is  in  the  form  of  snow  or  hoar  frost. 

The  reverse  process  is  similar.  Suppose  the  temperature 
of  snow  is  slowly  being  increased.    If  there  is  only  a  very 


282      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  ix,  172 

little  water  vapor  in  the  air  surrounding  it,  the  snow  evapo- 
rates into  water  vapor  without  first  melting.  On  the  other 
hand,  if  the  atmosphere  contains  an  abundance  of  water 
vapor,  the  snow  -does  not  evaporate  until  after  its  tempera- 
ture has  risen  above  the  freezing  point.  But  at  the  freezing 
point  the  snow  turns  into  water. 

The  gist  of  the  whole  matter  is  this :  If  the  water  vapor 
in  the  atmosphere  is  abundant,  precipitation  and  evapora- 
tion take  place  above  the  freezing  point ;  and  if  it  is  scarce, 
precipitation  and  evaporation  take  place  below  the  freezing 
point.  The  temperature  at  which  these  processes  begin 
depends  only  on  the  density  of  water  vapor  present  and  not 
at  all  upon  the  constitution  and  density  of  the  remainder  of 
the  atmosphere.  For  example,  snow  evaporates  (or  sub- 
limes) at  —35°  Fahrenheit  when  the  density  of  the  water 
vapor  surrounding  it  is  such  that  its  pressure  is  less  than 
0.00018  of  ordinary  atmospheric  pressure;  or,  if  this  is  the 
water- vapor  pressure  and  the  temperature  falls  below  ^35°, 
snow  is  precipitated.  Similarly,  water  evaporates  at .  80° 
Fahrenheit  if  the  pressure  of  the  water  above  it  is  less  than 
0.034  of  atmospheric  pressure;  or,  with  this  pressure  of 
water  vapor,  precipitation  occurs  if  the  temperature  falls 
below  80°.  This  explains  why  the  earth's  atmosphere  on 
the  whole  is  much  dryer  in  the  winter  than  it  is  in  the  summer. 

The  application  to  Mars  is  simple.  Suppose  its  polar 
caps  are  actually  due  to  snow  or  hoar  frost,  as  they  appear 
to  be.  The  fact  that  they  nearly  or  entirely  disappear  in 
the  long  summers  means  only  that  the  atmosphere  is  dry 
enough  for  evaporation  to  take  place  at  the  temperature 
which  prevails  on  the  planet.  If  the  temperature  of  Mars 
were  known,  the  amount  of  water  vapor  in  its  atmosphere 
could  be  computed  from  the  phenomena  of  the  polar  caps ;  and 
conversely,  if  the  amount  of  water  vapor  in  the  atmosphere 
of  Mars  were  known,  its  temperature  could  be  computed. 

Some  direct  considerations  on  the  possible  temperature 
of  Mars  have  been  given,  and  reference  has  been  made  to 


CH.  IX,  173]  THE   PLANETS  283 

the  possibility  of  determining  the  water  content  of  its 
atmosphere"  by  means  of  the  spectroscope.  There  is  an 
additional  hne  of  evidence  which  bears  on  the  question  in 
hand.  The  surface  of  the  planet  is  largely  of  a  brick-red 
color,  and  is  interpreted  as  being  in  a  desert  condition.  While 
there  are  dark  regions  which  have  been  supposed  possibly  to 
be  marshes,  there  are  certainly  no  large  bodies  of  water  on 
its  surface  comparable  to  the  oceans  and  seas  upon  the 
earth.  These  things  confirm  the  conclusion  that  water  is 
not  abundant  on  Mars  and  that  its  mean  temperature  may 
be  below  zero ;  but,  in  the  equatorial  regions  in  the  long  sum- 
mers, the  temperature  may  rise  for  a  considerable  time  even 
above  the  freezing  point. 

173.  The  Canals  of  Mars.  —  In  1877,  Schiaparelli,  an 
ItaUan  observer  of  Milan,  made  the  first  of  a  series  of  impor- 
tant discoveries  respecting  the  surface  markings  of  Mars. 
Although  he  had  only  a  modest  telescope  of  8.75  inches'  aper- 
ture, he  found  that  the  brick-red  regions,  which  had  been 
supposed  to  be  continental  areas,  were  crossed  and  recrossed 
by  many  straight,  dark,  greenish  streaks  which  always 
ended  in  the  darker  regions  known  as  "  seas."  These  streaks 
were  of  great  length,  extending  in  uniform  width  from  a  few 
-hundred  miles  up  to  three  or  four  thousand  miles.  While 
they  appeared  to  be  very  narrow,  they  must  have  been  at 
least  20  miles  across.  SchiapareUi  called  them  "  canali  " 
(channels),  which  was  translated  as  "  canals,"  a  designation 
unfortunately  too  suggestive,  for  they  have  no  analogy  to 
anything  on  the  earth.  When  a  number  of  them  intersect, 
there  is  generally  a  dark  spot  at  the  point  of  intersection 
which  is  called  a  "  lake."  Sometimes  a  number  of  them 
intersect  at  a  single  point ;  and,  according  to  Lowell,  the 
junctions  of  canals  are  always  surrounded  by  lakes,  while 
lakes  are  found  at  no  other  places. 

In  the  winter  of  1881-82  Mars  was  again  in  opposition, 
though  not  so  near  the  earth  as  in  1877.  SchiapareUi  not 
only  verified  his  earlier  observations,  but  he  also  found  the 


284       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  173 

remarkable  fact  that  a  number  of  the  canals  had  doubled; 
that  is,  that,  in  a  number  of  cases,  two  canals  ran  parallel 
to  each  other  at  a  distance  of  from  200  to  400  miles,  as  shown 
on  Lowell's  map  in  Fig.  106,  which  is  a  photograph  of  a 
globe  on  which  he  had  drawn  all  the  markings  he  had 
observed.     The  doubUng  was  found  to  depend  upon  the  sea^ 


Fig.  106.  —  Lowell's  map  of  Mars. 


sons  and  to  develop  with  great  rapidity  when  the  sun  was 
at  the  Martian  equinox. 

The  history  of  the  observations  of  the  markings  of  Mars 
since  the  time  of  SchiaparelU  is  filled  with  the  most  remark- 
able contradictions.  The  observations  of  the  keen-eyed 
Italian  have  been  confirmed  by  a  number  of  other  astrono- 
mers, among  whom  may  be  mentioned  Perrotin  and  ThoUon, 
of  Nice,  Williams,  of  England,  W.  H.  Pickering,  of  Harvard, 
and  especially  Lowell,  who  has  a  large  24-inch  telescope 


CH.  IX,  174] 


THE   PLANETS 


285 


favorably  located  at  Flagstaff,  Arizona.  On  the  other  hand, 
many  of  the  foremost  observers  working  with  the  very  lar- 
gest telescopes,  such  as  Antoniadi,  with  the  32.75-inch  Meu- 
don  refractor,  the  Lick  observers,  with  the  great  36-inch  ' 
telescope,  Barnard,  with  the  40-inch  Yerkes  telescope,  and 
Hale,  with  the  60-inch  reflector  of  the  Solar  Observatory  at. 
Mt.  Wilson,  California,  have  been  entirely  unable  to  see  the 
canals.  This  does  not  mean  that  they  have  not  seen  mark- 
ings on  Mars,  for  they 
have  observed  many  of 
them ;  but  they  do  not 
find  the  narrow,  straight 
lines  observed  by  Schia- 
parelU,  Lowell,  and 
others.  In  Fig.  107  four 
views  of  Mars  are  shown 
as  seen  by  Barnard  with 
the  great  telescope  of  the 
Lick  Observatory,  and 
Fig.  108  is  a  photograph 
made  with  the  60-inch 
reflecting  telescope  of  the 
Mt.  Wilson  Solar  Ob- 
servatory. In  the  nlidst 
of  these  conflicting  results 

it  is  difficult  to  draw  any  certain  conclusion;  but  it  must 
be  remembered  in  considering  such  a  subject  that  reliable 
positive  evidence  ought  to  outweigh  a  large  amount  of  nega- 
tive evidence. 

174.  Explanations  of  the  Canals  of  Mars.  —  The  explana- 
tions of  the  canals  of  Mars  have  been  extremely  varied. 
Many  astronomers  believe  they  are  illusions  of  some  sort. 
They  think  the  eye  in  some  way  integrates  the  numerous 
faint  markings  which  certainly  exist  on  Mars  into  straight 
lines  and  geometrical  figures.  The  experiments  of  Maunder 
and  Evans  and  the  more  recent  ones  of  Newcomb  of  having 


Fig.  107.  —  Drawings  of  Mars  in  1894  by 
Barnard  at  the  Lick  Observatory. 


286       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  ix,  174 

a  number  of  persons  make  drawings  of  what  they  could  see 
on  a  disk  covered  with  irregular  marks  and  held  slightly 
beyond  the  limits  of  distinct  vision,  strikingly  confirm  this 
conclusion.  Antoniadi  states  in  the  most  unequivocal  terms 
that  the  observations  of  Mars  at  the  opposition  of  1909  give 


Fig.  108.  —  Photograph  of  Mars  (the  60-inch  reflector  of  the  Mt.  Wilson 
Solar  Observatory) . 

to  the  theory  of  the  objective  existence  of  canals  on  Mars 
an  unanswerable  confutation.  Other  astronomers  hold  that 
such  a  network  of  markings  on  a  planet  whose  surface  is 
certainly  somewhat  uneven  is  inherently  improbable,  and 
should  not  be  accepted  without  the  most  conclusive  evidence. 
At  the  other  extreme  stands  Lowell,  who  maintains  that 


CH.  IX,  174]  THE   PLANETS  287 

not  only  are  the  canals  real  but  that  they  prove  the  existence 
on  the  planet  of  highly  intelligent  beings.  He  argues  for 
the  reahty  of  the  canals  on  the  ground  that  they  always 
appear  at  well-defined  positions  on  the  planet  and  that  they 
change  in  a  systematic  way  with  the  seasons.  He  argues  that 
they  are  artificial  because  they  always  run  along  the  arcs  of 
great  circles,  because  several  of  them  sometimes  cross  at  a 
point  with  the  utmost  precision,  and  because  in  many  cases 
two  of  them  run  perfectly  parallel  for  more  than  a  thousand 
miles.  Obviously  this  remarkable -regularity  could  not  be 
the  result  of  such  processes  as  the  erosion  of  rivers  or  the 
cracking  of  the  surface. 

W.  H.  Pickering  first  suggested  that  the  canals  may  be 
due  to  vegetation,  and  Lowell's  theory  is  an  elaboration  of 
this  idea.  Lowell  believes  the  streaks,  known  as  canals, 
are  strips  of  vegetation  20  or  more  miles  wide,  which  grow 
on  a  region  irrigated  by  lateral  ditches  from  a  large  central 
canal.  This  explains  their  seasonal  character.  Moreover, 
he  finds  the  streaks  first  developing  near  the  dark  (marshy  ?) 
regions  and  extending  gradually  out  from  them  even  across 
the  equator  of  the  planet  to  regions  having  the  opposite  sea- 
son. The  explanation  given  for  this  phenomenon  is  that 
when  the  snow  of  the  polar  caps  melts,  the  resulting  water 
first  collects  in  the  marshes  and  is  led  thence  out  into  the 
waterways  which  extend  through  the  centers  of  the  canals. 
The  observations  of  Lowell  show_  that,  according  to  his 
explanation,  water  must  flow  along  the  canals  at  the  rate 
of  2.1  miles  per  hour.  He  infers  from  the  elaborate  system 
of  irrigated  regions  that  Mars  is  inhabited  by  creatures 
possessing  a  high  order  of  intelligence. 

Although  Lowell's  theory  seems  highly  improbable  and  may 
be  altogether  wrong,  life  may  nevertheless  exist  upon  Mars. 
But  if  there  is  hfe  on  this  planet,  the  creatures  which  inhabit 
it  must  be  very  different  physically  from  those  on  the  earth, 
because  it  would  be  necessary  for  them  to  be  adapted  to  an 
entirely  different  environment.     On  Mars  the  surface  gravity 


288       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  174 

is  less  than  on  the  earth,  the  hght  and  heat  received  from 
the  sun  are  less  and  the  temperature  is  probably  far  lower, 
the  atmosphere  is  much  less  abundant,  and  it  may  be  quite 
different  in  constitution,  and  the  seasonal  changes  are 
nearly  twice  as  long.  The  plants  and  animals  which  in- 
habit the  earth  are  more  or  less  perfectly  adapted  to  the 
conditions  existing  on  its'  surface,  and  the  conditions  have 
not  been  made  to  fit  them,  as  was  once  generally  believed. 
Similarly,  life  on  other  planets  must  be  adapted  to  the 
environment  in  which  it  is  placed  or  it  would  shortly  perish. 

Further,  if  Mars  or  any  other  world  is  inhabited,  there  is 
no  reason  to  suppose  that  its  highest  intelhgence  has  reached 
the  precise  stage  attained  by  the  hiunan  race.  The  most 
intelligent  creatures  on  another  planet  may  be  in  the  condi- 
tion corresponding  to  that  in  which  our  ancestors  were  when 
they  lived  in  caves  and  ate  uncooked  food ;  or,  millions  of 
years  ago  they  may  have  passed  through  the  stage  of  strife 
and  deadly  competition  in  which  the  human  race  is  to-day. 

It  is  a  curious  fact  that  those  who  know  but  little  about 
astronomy  are  nearly  always  very  much  interested  iri  the 
question  whether  other  worlds  are  inhabited,  while  as  a  rule 
astronomers  who  devote  their  whole  lives  to  the  subject 
scarcely  give  the  question  of  the  habitability  of  other  planets 
a  thought.  Astronomers  are  doubtless  influenced  by  the 
knowledge  that  such  speculations  can  scarcely  lead  to  cer- 
tainty, and  they  are  .deeply  impressed  by  the  fundamental 
laws  which  they  find  operating  in  the  universe.  Nevertheless, 
there  seems  to  be  no  good  reason  why  we  should  not  now 
and  then  consider  the  question  of  the  existence  of  life,  not 
only  on  the  other  planets  of  the  solar  system,  but  also  on  the 
millions  of  planets. that  possibly  circulate  around  other  suns. 
Such  speculations  help  to  enlarge  our  mental  horizon  and 
to  give  us  a  better  perspective  in  contemplating  the  origin 
and  destiny  of  the  human  race,  but  we  should  never  forget 
that  they  are  speculations. 


CH.  IX,  175]  THE   PLANETS  289 

III.   Jupiter 

175.  Jupiterls  Satellite  System.  —  The  first  objects  dis- 
covered by  Galileo  when  he  pointed  his  little  telescope  to 
the  sky  in  1610  were  the  four  brightest  moons  of  Jupiter. 
They  are  barely  beyond  the  limits  of  visibility  without  optical 
aid  and,  indeed,  could  be  seen  with  the  unaided  eye  if  they 
\^^ere  not  obscured  by  the  dazzling  rays  of  Jupiter.  No  other 
satellite  of  Jupiter  was  discovered  until  1892,  when  Barnard, 
then  at  the  Lick  Observatory,  caught  a  glimpse  of  a  fifth 
one  very  close  to  the  planet.  It  is  so  small  and  so  buried 
in  the  rays  of  the  neighboring  bpUiant  planet  that  it  can 
be  seen  only  by  experienced  observers  with  the  aid  of  the 
most  powerful  telescopes  in  the  world. 

Early  in  1905  Perrine  found  by  photography  that  Jupiter 
has  still  two  more  satellites  which  are  more  remote  from  the 
planet  than  those  previously  known.  Their  distances  from 
Jupiter  are  both  about  7,000,000  miles  and  their  periods  of 
revolution  are  about  0.75  of  a  year.  The  eccentricities  .of 
their  orbits  are  considerable  and  their  paths  actually  loop 
through  one  another.  The  mutual  inclination  of  their 
orbits  is  28°  and  they  do  not  pass  nearer  than  2,000,000 
miles  of  each  other. 

The  seven  satellites  so  far  enumerated  revolve  around 
Jupiter  from  west  to  east,  but  two  more  have  been  dis- 
covered whose  motion  is  in  the  opposite  direction.  The 
eighth  was  found  by  Melotte,  at  Greenwich,  England,  in 
January,  1908.  It  revolves  around  Jupiter  at  a  mean 
distance  of  approximately  14,000,000  miles  in  a  period 
of  about  740  days.  Its  orbit  is  inclined  to  Jupiter's  equa- 
tor by  about  28°.  The  ninth  was  discovered  by  S.  B. 
Nicholson,  in  July,  1914,  at  the  Lick  Observatory.  Its 
mean  distance  from  Jupiter  is  about  15,400,000  miles  and  its 
period  is  nearly  3  years.  These  remote  satelhtes  are  very 
small  and  faint,  the  ninth  being  of  the  nineteenth  magni- 
tude, and  the  eighth  about  one  magnitude  brighter. 


290       AN    INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  175 

The  first  four  satellites  discovered  are  numbered  I,  II, 
III,  IV  in  the  order  of  their  distance  from  Jupiter.  The 
fifth,  although  it  is  very  close  to  Jupiter,  was  given  the 
number  V.  The  orbits  of  these  five  satellites,  shown  in 
Fig.  109,  are  nearly  circular  and  lie  in  the  plane  of  Jupiter's 
equator.  The  four  larger  satellites  are  of  considerable 
dimensions  and  their  diameters  have  been  determined  by 
Barnard,  the  results  being  given  in  the  following  table :  ■ 

Table  VII 


Satellite 

Distance  from 
Center  of  Jupiter 

Period  of 
Revolution 

Diameter 

V  (Unnamed)  ; 

112,500  mi. 

Od.  llh.  57m. 

about   100  mi. 

I  (lo)     .     . 

261,000  mi. 

Id.  18h.  28m. 

2452  mi. 

II  (Europa)      . 

415,000  mi. 

3d.  13h.  14m. 

2045  mi. 

Ill  (Ganymede) 

664,000  mi. 

7d.    3h.  43m. 

3558  mi 

IV  (Callisto) 

1,167,000  mi. 

16d.  16h.  32m. 

3345  mi. 

VI  (Unnamed)  . 

7,300,000  mi. 

about  266  days 

small 

VII  (Unnamed)  . 

7,500,000  mi. 

about  277  days 

smaU 

VIII  (Unnamed)  . 

14,000,000  ±  mi. 

about  740  days 

very  small 

IX  (Unname^)  . 

15,400,000  ±  mi. 

nearly  3  years 

very  small 

176.  Markings  on  Jupiter's  Satellites.  —  The  great  dis- 
tance of  Jupiter  makes  it  difficult  to  detect  any  but  large 
and  distinctly  colored  markings  on  its  satellites.  In  1890 
Barnard  found  satellite  I  to  be  elongated  parallel  to  the 
equator  of  Jupiter  when  transiting  its  darker  portions  and 
elongated,  or  double,  in  the  opposite  direction  when  passing 
over  its  brighter  parts.  He  interpreted  this  as  meaning  that 
the  poles  of  the  satellite  are  dark  and  that  the  equatorial 
belt  is  light  colored.  The  accompanying  drawing,  Fig.  110; 
showing  the  satellite  transiting  a  light  region  above  and  a 
dark  one  below,  exhibits  the  observed  appearance  at  the 
left  and  the  probable  actual  condition  at  the  right.  When 
held  at  some  distance  from  the  eye,  the  two  appear  the 
same. 


CH.  IX,  177] 


THE   PLANETS 


291 


Some  observers  have  thought  that  satelUtes  III  and  lY  are 
somewhat  elliptical  in  shape,   but  Barnard  has  observed 
them  repeatedly  with 
the    great   Lick  and  — "^ 

Yerkes  telescopes  and 
has  been  quite  unable 
to  detect  in  them  any 
departures  from  strict 
sphericity.  Various 
markings  have  been 
at  times  observed  on 
the  satellites,  and 
Douglas  inferred  from 
his  observations  of 
satelUte  III  that  its 
period  of  rotation  is 
about  7  hours.  "  At 
present  these  are  mat- 
ters of  speculation.  ^ 

177.  Discovery  of  the  Finite  Velocity  of  Light.  —  A  very 
important  discovery  was  made  in  connection  with  observa- 
tions of  Jupiter's  satelhtes.     The  periods  of  revolution  of  the 


Fig.   109.- 


-  Orbits  of  first  four  satellites  of 
•Jupiter. 


■^^^^^^m 


Fig.  110.  —  Barnard's  drawings 
of  Jupiter's  satellite  I. 


four  largest  satellites  naturally 
were  determined  when  Jupiter 
w-as  in  opposition,  and  therefore 
nearest  the  earth.  Since  the 
satellites  are  in  the  plane  of 
Jupiter's  equator,  which  is  only 
slightly  incHned  to  the  ecliptic, 
they  are  eclipsed  when  they 
pass  behind  Jupiter.  From  their 
periods  of  revolution  the  times 
at  which  they  will  be  eclipsed 
can  be  predicted. 

Suppose  the  periods  of  revo- 
lution of  the  satellites  and  the 


292       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  177 

times  at  which  they  are  eclipsed  are  determined  when  the 
earth  is  in  the  vicinity  of  Ei,  Fig.  111.  Six  months  later, 
when  the  earth  has  arrived  at  E2,  its  distance  from  Jupiter 
is  greater  by  approximately  the  diameter  of  the  earth's 
orbit,  and  then  the  eclipses  of  the  satellites  are  found  to 
be  behind  their  predicted  times  by  the  time  required  for 
light  to  travel  across  the  earth's  orbit.  From  such  obser- 
vations, in  1675,  the  Danish  astronomer  Romer  inferred  that 
it  takes  light  600  seconds  to  travel  a  distance  equal  to  that 
from  the  sun  to  the  earth.  Later  observations  have  shown 
that  the  correct  time  is  498.58  seconds.     When  the  distance 

\ 


^'/ 


5# 


Fia.  111.  —  Discovery  of  velocity  of  light  from  eclipses  of  Jupiter's  satellites. 

from  the  earth  to  the  sun  has  been  determined  by  independ- 
ent means,  the  velocity  of  light  can  be  found  from  this 
interval,  which  is  called  the  light  equation. 

At  the  present  time  the  velocity  of  light  can  be  deter- 
mined much  more  accurately  by  physical  experiments  on 
the  surface  of  the  earth  than  it  can  irom  observations  of 
Jupiter's  satellites.  The  work  of  Fizeau,  Michelson,  and 
Newcomb  shows  that  it  is  very  approximately  186,324  miles 
per  second.  From  this  velocity  and  the  light  equation  of 
498.58  seconds,  the  distance  to  the  sun  can  be  compulsed. 

178.  The  Rotation  of  Jupiter.  —  The  surface  of  Jupiter 
is  covered  with  a  great  number  of  semi-permanent  mark- 
ings from  which  its  rotation  can  be  determined.     The  period 


CH.  IX,  179]  THE   PLANETS  293 

of  rotation  for  spots  near  the  equator  has  been  found  to  be 
about  9  hrs.  and  50  m.,  and  for  those  in  higher  latitudes  about 
9  hrs.  and  57  m.,  with  an  average  of  9  hrs.  and  54  m. ;  that 
is,  between  the  equatorial  zone  and  high  latitudes  there  is  a 
difference  in  the  period  of  about  ^  of  the  whole  period. 
In  85  rotations  the  equator  gains  a  rotation  on  the  higher 
latitudes.  Moreover,  as  Barnard  has  found,  the  rates  of 
rotation  in  corresponding  northern  and  southern  latitudes 
are  quite  different  in  several  zones. 

The  circumference  of  Jupiter  is  nearly  300,000  miles,  and 
it  follows  from  this  and  its  rate  of  rotation  that  the  motion 
at  its  equator  is  about  30,000  miles  per  hour.  Consequently, 
if  two  spots  whose  periods  of  rotation  differ  by  7  minutes 
were  both  near  the  equator,  they  would  pass  each  other 
with  the  relative  speed  of  30,000  -4-  85  =  353  miles  per 
hour.  Though  spots  whose  periods  differ  by  7  minutes  are 
probably  in  no  case  in  approximately  the  same  latitude,  yet 
they  must  have  large  relative  motions.  Compare  these 
results  with  the  speed  of  from  70  to  100  miles  per  hour  with 
which  tornadoes  sweep  along  the  surface  of  the  earth. 

The  fact  that  the  equatorial  belt  of  Jupiter  rotates  in  a 
shorter  period  than  its  higher  latitudes  is  a  most  remarkable 
phenomenon.  If  it  were  an  isolated  case,  one  would  natu- 
rally suppose  that  the  pecuKarity  was  due  to  irregularities 
of  motion  inherited  from  the  time  of  its  origin.  Such  cur- 
rents in  a  body  in  a  fluid  condition  would  be  destroyed  by 
friction  only  very  slowly;  but  the  same  phenomenon  is 
also  found  in  the  case  of  Saturn  and  the  sun.  It  can  hardly 
be  supposed  that  the  three  are  mere  coincidences.  If  they 
are  not,  the  impHcation  is  that  these  pecuHarities  of 
rotation  have  been  produced  by  similar  causes.  It  has 
been  suggested,  as  will  be  explained  in  Arts.  253,  254,  that 
the  cause  may  be  the  impacts  of  circulating  meteors  or  other 
material. 

179.  Surface  Markings  of  Jupiter.  —  The  characteristic 
markings  of  Jupiter  are  a  series  of  conspicuous  dark  and 


294       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  179 


bright  belts  which  stretch  around  the  planet  parallel  to  its 
equator  as  shown  in  Figs.  112,  113,  and  114.  The  central 
equatorial  belt  is  usually  very  light  and  about  10,000  miles 
wide ;  on  each  side  is  a  belt  of  reddish-brown  color  generally 
of  about  the  same  width.  Several  other  alternately  light 
and  dark  belts  can  be  made  out  in  higher  latitudes,  though 
not  as  distinctly  as  the  equatorial  belts,  partly,  at  least, 
because  they  are  observed  obliquely.  The  belts  vary  con- 
siderably in  width  from  year  to   year   as  the   drawings, 

Fig.  114,  by  Hough 
show.  On  the  whole, 
the  southern  dark  belt 
of  Jupiter  is  wider 
and  more  conspicuous 
than  the  northern 
one. 

A  goo'd  telescope 
under  favorable  at- 
mospheric conditions 
reveals  in  the  belts 
many  details  which 
continually  change  as 
though  what  we  see 
is  cloudhke  in  struc- 
ture. In  fact,  it  follows 
from  the  low  mean  density  of  the  planet  and  the  almost 
certain  central  condensation  that  its  exterior  parts,  to  a 
depth  of  many  thousands  of  miles,  must  have  a  very  low 
density ;  and  it  is  improbable  that  anything  which  is  visible 
from  the  earth  approaches  the  solid  state. 

Dark  spots  often  appear  on  Jupiter,  especially  in  the  north- 
ern hemisphere,  which  gradually  turn  red  and  finally  van- 
ish. The  most  remarkable  and  permanent  spot  so  far  known 
appeared  in  1878  just  beneath  the  southern  red  belt.  When 
first  discovered  it  was  a  pinkish  oval  about  7000  miles  across 
in  the  direction  perpendicular  to  the  equator,  and  about 


Fig.  112.  —  Jupiter,  Sept.  7,  1913  (Barnard). 


CH.  IX,  179] 


THE   PLANETS 


295 


30,000  miles  long  parallel  to  the  equator.  In  a  year  it  had 
changed  to  a  bright  red  color  and  was  by  far  the  most  con- 
spicuous object  on  the  planet.  It  has  since  then  been  known 
as  "  the  great  red  spot,"  but  it  has  undergone  many  changes, 
both  in  color  and  brightness.  At  the  present  time  it  has 
become  rather  inconspicuous,  and  the  material  of  which  it  is 


Fig.   113.  —  Photographs  of  Jupiter  (E.  C.  Slipher,  Lowell  Observatory). 

composed  seems  to  be  sinking  back  beneath  the  vapors  which 
surround  the  planet. 

A  very  remarkable  thing  in  connection  with  the  red  spot 
was  that  its  period  of  rotation  increased  7  seconds  the  first 
eight  years  following  its  discoyery,  but  it  has  remained  essen- 
tially constant  since  that  time.  Possibly  the  increase  in 
period  of  rotation  of  the  red  spot,  which  was  somewhat 
longer  than  that  of  the  surrounding  material  which  continu- 
ally flowed  by  it,  was  due  to  its  being  elevated  so  that  its 
distance  from  the  axis  of  rotation  of  the  planet  was  increased. 
Under  these  conditions  the  rate  of  rotation  would  be  reduced 


296      AN  INTRODUCTION  TO  ASTRONOMY    [ch.  ix,  179 


in  harmony  with  the  principle  of  the  conservation  of  moment 
of  momentum  (Art.  45).  At  any  rate,  changes  in  rotation 
are  always  accompanied  by  considerable  changes  in  color 
and  visibility  of  the  parts  affected. 

180.  The  Physical  Condition  and  Seasonal  Changes  of 
Jupiter.  —  In  considering  the  physical  condition  of  Jupiter 
it  should  be  remembered  that  it  has  the  low  average  density 

of  1.25  on  the  water 
standard,  that  its  surface 
markings  are  not  perma- 
nent, and  that  there  are 
violent  relative  motions 
of  its  visible  parts.  All 
these  things  indicate  that 
Jupiter  is  largely  gaseous 
near  its  surface. 

The  surface  gravity  of 
Jupiter  is  2.52  times  that 
of  the  earth,  and  this 
produces  great  pressures 
in  its  atmosphere  at 
moderate  depths.  These 
pressures  are  sustained 
by  the  expansive  tenden- 
cies of  the  interior  gases 
which  may  be  composed 
of  light  elements,  or 
which  may  have  high 
temperatures.  It  has 
sometimes  been  supposed  that  the  surface  of  Jupiter  is  very 
hot  and  that  it  is  self-luminous,  but  such  cannot  be  the  case, 
for  the  shadows  cast  on  the  planet  by  the  satellites  are 
perfectly  black,  and  when  a  satellite  passes  into  the  shadow 
of  Jupiter  it  becomes  absolutely  invisible. 

In  conclusion,  we  shall  probably  not  be  far  from  the 
truth  if  we  infer  that  Jupiter  is  still  in  an  early  stage  of  its 


Fig.  1 14.  —  Drawings  of  Jupiter  show- 
ing variations  in  widths  of  dark  belts 
(Hough). 


-  -^;;:s»s;;«(iHE3i.*(;Ji> 


CH.  IX,  181]  THE   PLANETS  297 

evolution,  rather  than  far  advanced  like  the  terrestrial 
planets,  that  it  contains  enormous  volumes  of  gases  which 
are  in  rapid  circulation  both  along  and  perpendicular  to  its 
surface,  and  that  possibly  the  energy  of  its  internal  fires 
gives  rise  to  violent  motions. 

The  eccentricity  of  Jupiter's  orbit  is  very  small  and  the 
plane  of  its  equator  is  inclined  only  3°  5'  to  the  plane  of  its 
orbit.  The  factors  which  produce  seasonal  changes  are, 
therefore,  unimportant  in  the  case  of  this  planet.  Its  dis- 
tance from  the  sim  is  so  great  that  it  receives  per  unit  area 
only  -^  as  much  hght  and'  heat  as  is  received  by  the  earth ; 
and,  consequently,  its  surface  must  be  cold  unless  it  is 
warmed  by  internal  heat. 

IV.   Satukn 

181.  Saturn's  Satellite  System.  —  Saturn,  like  Jupiter, 
has  9  satelHtes.  The  largest  one  was  discovered  by  Huyghens 
in  1655,  then  four  more  were  found  by  J.  D.  Cassini  between 
1671  and  1684,  two  by  Wilham  Herschel  in  1789,  one  by 
G.  P.  Bond  and  Lassell  in  1848,  and  the  ninth  by  W.  H. 
Pickering  in  1899^  Pickering  suspected  the  existence  of 
a  tenth  in  1905,  but  the  supposed  discovery  has  not  been 
confirmed. 

Saturn  is  so  remote  that  the  dimensions  of  its  sateUites  are 
only  roughly  known  from  their  apparent  brightness.  All 
their  masses  are  unknown  except  that  of  Titan,  which,  from 
its  perturbation  of  its  neighboring  satellite  Hyperion,  was 
found  by  Hill  to  be  ttTt  that  of  Saturn.  The  7  satelUtes 
which  are  nearest  to  Saturn  revolve  sensibly  in  the  plane 
of  its  equator,  while  the  orbit  of  the  eighth,  Japetus,  is 
inclined  about  10°,  and  that  of  the  ninth  about  20°. 

When  the  eighth  satellite,  Japetus,  is  on  the  western  side 
of  Saturn  it  always  appears  considerably  brighter  than  when 
it  is  on  the  eastern  side.  This  difference  in  brightness  is 
undoubtedly  due  to  the  fact  that  this  satellite,  like  the  moon, 
always  has  the  same  side  toward  the  planet  around  which 


298       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  ix,  181 

it  revolves,  and  that  its  two  sides  reflect  light  very  unequally. 
Similar,  but  less  marked,  phenomena  have  been  observed  by 
Lowell  and  E.  C.  Shpher  in  connection  with  the  first  two 
satellites,  and  the  explanation  is  the  same  as  in  the  case  of 
Japetus. 

Table  VIII  gives  the  Ust  of  Saturn's  satelUtes,  together 
with  their  mean  distances  from  its  center,  their  periods,  and 
their  approximate  diameters.  It  will  be  observed  that  an 
enormous  gap  separates  the  first  eight  from  the  ninth. 

Figure  115  gives  to  scale  the  orbits  of  Saturn's  satellites, 
with  the  exception  of  the  ninth,  which  is  too  remote  to  be 
shown.  The  eight  satellites  revolve  around  Saturn  from 
west  to  east,  the  direction  in  which  it  rotates,  but  the  ninth, 
Hke  the  eighth  and  ninth  satelUtes  of  Jupiter,  revolves  in 
the  retrograde  direction.  This  satellite  was  the  first  object 
discovered  in  the  solar  system  having  retrograde  motion, 
and  it  aroused  great  interest.  These  retrograde  revolutions 
have  a  fundamental  bearing  on  the  question  of  the  origin 
of  the  satellite  systems. 

Table  VIII 


Satellite 


Distance  prom 

Center  of 

Saturn 


Period  op 
Revolution 


Diameter 


I  (Mimas) 
II  (Bnceladus) 

III  (Tethys)      . 

IV  (Dione)  .     . 
V  (Rhea)    .     . 

VI  (Titan)  .  . 
Vll  (Hyperion) 
Vlll  (Japetus)  . 
IX  (Phoebe)   . 


117,000  mi. 

157,000  mi. 

186,000  mi. 

238,000  mi. 

332,000  mi, 

771,000  mi, 

934,000  mi, 
2,225,000  mi, 
7,996,000  mi 


Od.  22h.  37m. 

1  8  53 

1  21  18 

2  17  41 
4  12  25 

15  22  41 

21  6 

79  7 

546  12 


39 

54 

0 


about 
about 
about 
about 
about 
about 
about 
about 
about 


600  mi, 

800  mi, 
1200  mi, 
1100  mi, 
1500  mi, 
3000  mi 

500  mi 
2000  mi 

200  mi 


The  question  may  be  asked  why  the  remote  satellites  of 
both  Jupiter  and  Saturn  revolve  in  the  retrograde  direction. 
This  question  cannot  be  answered  with   certainty  at  the 


CH.  IX,  182] 


THE   PLANETS 


299 


present  time.  But  it  is  certain  that  the  farther  a  satellite 
is  from  a  planet,  the  less  securely  is  it  held  under  the  gravita- 
tive  control  of  its  primary ;  and  there  is  a  distance  beyond 
which  a  satellite  cannot  permanently  revolve  because  it 
would  abandon 
the      planet      in  osgE°£-££ro. 

obedience  to  the 
greater  attraction 
of  the  sun.  A 
mathematical  dis- 
cussion of  the 
problem  shows 
that,  at  a  given 
distance  from  a 
planet,  motion  in 
the  retrograde  di- 
rection is  much 
more  stable  than 
in  the  forward 
direction;  and 
consequently,  out 
near  the  region 
where  instabiUty  begins,  it  would  be  expected  that  only 
retrograde  satelUtes  would  be  found.  The  orbit  of  the  ninth 
satelhte  of  Saturn  is  in  the  region  of  stabihty  even  for  direct 
motion;  but  Jupiter's  eighth  and  ninth  satellites  would 
both  have  unstable  orbits  if  they  revolved  in  the  forward 
direction  at  the  same  distances  from  Jupiter. 

182.  Saturn's  Ring  System.  —  Saturn  is  distinguished  from 
all  the  other  planets  by  three  wide,  thin  rings  which  extend 
around  it  in  the  plane  of  its  equator.  They  were  first  seen 
by  Galileo  in  1610,  but  their  true  character  was  not  known 
until  the  observations  of  Huyghens  in  1655.  The  dimensions 
of  Saturn  and  its  ring  system  according  to  the  extensive 
measurements  of  Barnard  are  given  in  Table  IX. 


Fig.   115.  —  Orbit  of  Saturn's  satellites. 


300       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  182 


Table  IX 


EquatoriaL  radius  of  Saturn     ...          .     . 

.  38,235  miles 

Center  of  Saturn  to  inner  edge  of  crape  ring 

.  44,100  miles 

Center  of  Saturn  to  inner  edge  of  bright  ring 

.  55,000  miles 

Center  of  Saturn  to  outer  edge  of  bright  ring 

.  73,000  miles 

Center  of  Saturn  to  inner  edge  of  outer  ring 

.  75,240  miles 

Center  of  Saturn  to  outer  edge  of  outer  ring 

.  B6,300  miles 

The  distance  from  the  surface  of  Saturn  to  the  inner  edge 
of  the  thin,  faint  ring,  known  as  "  the  crape  ring,"  is  nearly 
6000  miles.  The  width  of  the  crape  ring  is  about  11,000 
miles.  Outside  of  the  crape  ring  is  the  main  bright  ring, 
whose  width  is  about  18,000  miles.     Its  brightness  increases 


Fig.  116.  —  Saturn  with  rings  tilted  at  greatest  angle  (drawing  by  Barnard). 


from  its  junction  with  the  crape  ring  outward  nearly  to  its 
outer  margin.  At  its  brightest  place  it  is  as  luminous  as  the 
planet  itself.  Beyond  the  main  bright  ring  there  is  a  dark 
gap  about  2200  miles  across.  It  is  known  as  "  Cassini's 
Division  "  because  it  was  first  observed  by  Cassini.  Outside 
of  this  dark  space  is  the  outer  bright  ring  with  a  width  of 


CH.  IX,  182]  THE   PLANETS  301 

about  11,000  miles.    The  distance  across  the  whole  ring 
system  from  one  side  to  the  other  is  about  172,600  miles. 

The  rings  of  Saturn  are  inclined  about  27°  to  the  plane  of 
the  planet's  orbit  and  about  28°  to  the  plane  of  the  echptic. 
Consequently,  they  are  observed  from  the  earth  at  a  great 
variety  of  angles.  When  their  inclination  is  high,  Saturn 
and  its  ring  system  present  through  a  good  telescope  one  of 
the  finest  sights  in  the  heavens,  as  is  evident  from   Figs. 


Fig.  117.  —  Saturn.     Photographed  Nov.  19,  1911,  with  the  60-inch  telescope 
of  the  Mount  Wilson  Solar  Observatory. 

jlQ  and  117.  When  their  plane  passes  through  the  earth, 
they  appear  to  be  a  very  thin  hne  and  even  entirely  disap- 
pear from  view  for  a  few  hours,  as  Barnard  found  when 
observing  them  with  the  great  40-inch  telescope  in  1907. 
It  follows  that  the  rings  must  be  very  thin,  their  thickness 
probably  not  exceeding  50  miles .  When  the  rings  were  nearly 
edgewise  to  the  earth,  Barnard  could  see  them  faintly ;  but 
the  places  which  are  entirely  vacant  when  they  are  highly 
incUned  to  the  earth,  were  found  to  be  brighter  than  the  places 
where  the  rings  are  really  briUiant   (Fig.   118).     Barnard 


302      AN  INTRODUCTION  TO  ASTRONOMY    [ch.  ix,  182 

made  the  suggestion  that  this  appearance  is  due  to  the  fact 
that  Ught  shining  from  the  sun  through  the  open  regions  is 
reflected  back  from  the  interior  edges  of  the  denser  parts  of 
the  rings. 

183.  The  Constitution  of  Saturn's  Rings. —  The  bright 
rings  of  Saturn  have  the  same  appearance  of  solidity  and 
continuity  as  the  planet  itself.  It  was  generally  believed 
until  about  a  century  ago  that  they  were  sohd  or  fluid.  Yet 
since  1715,  when  J.  Cassini  first  mentioned  the  possibility, 
it  has  frequently  been  suggested  that  the  rings  may  be  simply 


Fig.  118.  —  Rings  of  Saturn,  December  12,  1907  (drawing  by  Barnard). 

swarms  of  meteors,  or  exceedingly  minute  satellites,  revolv- 
ing around  the  planet  in  the  plane  of  its  equator.  Such 
smaU  bodies  would  exert  only  neghgible  gravitational  influ- 
ences upon  one  another,  and  their  orbits  would  be  sensibly 
independent  of  one  another  except  for  colhsions. 

The  meteoric  theory  of  xhe  constitution  of  Saturn's  rings 
was  first  rendered  probable  by  Laplace,  who  showed  that 
a  symmetrical,  solid  ring  would  be  dynamically  unstable. 
That  is,  solid  rings  would  be  something  Uke  spans  of  enormous 
bridges,  whose  ends  do  not  rest  upon  the  planet  but  upon 
other  portions  of  the  rings.  They  would  have  to  be  com- 
posed of  inconceivably  strong  material  to  withstand  the 


CH.  IX,  183]  THE   PLANETS  303 

strains  due  to  their  motion  and  the  gravitational  forces  to 
which  they  would  be  subjected.  In  1857,  Clerk-Maxwell 
proved  from  dynamical  considerations  that  the  rings  could 
be  neither  sohd  nor  fluid,  and  that  they  were,  therefore,  com- 
posed of  small  independent  particles.  Now,  if  they  are 
meteoric,  those  parts  which  are  nearest  the  planet  must 
move  fastest,  just  as  those  planets  which  are  nearest  the  sun 
move  fastest ;  while,  if  they  are  solid,  the  opposite  must  be 
the  case.  In  1895,  Keeler  showed  by  line-of-sight  obser- 
vations with  the  spectroscope  (Art.  226)  that  the  inner  parts 
not  only  move  fastest,  but  that  all  parts  move  precisely 
as  they  would  move  if  they  were  made  up  of  totally  discon- 
nected particles,  the  innermost  particles  of  the  crape  ring 
performing  their  revolution  in  about  5  hours,  while  the  outer- 
most particles  of  the  outer  bright  ring  require  137  hours  to 
complete  a  revolution.  Moreover,  Barnard  found  that  they 
do  not  cast  perfectly  black  shadows,  for  he  saw  Japetus 
faintly  illuminated  by  the  rays  of  the  sun  which  filtered 
through  the  ring.  Hence  it  may  be  considered  as  firmly 
established  that  the  rings  of  Saturn  are  swarms  of  metfeors. 
Rings  are  strange  substitutes  for  satellites,  but  a  prob- 
able explanation  of  their  existence  in  place  of  satellites  is  at 
hand.  A  planet  exerts  tidal  strains  upon  satelHtes  in  its 
vicinity,  and  these  tendencies  to  rupture  increase  very 
rapidly  as  the  distance  of  the  satellite  decreases.  In  1848, 
Roche  proved  that  these  tidal  forces  would  break  up  a  fluid 
satellite  of  the  same  density  as  the  planet  around  which^t 
revolved  if  its  distance  were  less  than  2.44  . . .  radii  of 
the  planet.  The  Hmit  would  be  less  for  denser  satellites, 
and  a  little  less  for  solid  satellites,  but  not  much  less  if  they 
were  of  large  dimensions.  It  is  seen  from  the  numbers 
in  Table  IX,  or  from  Fig.  116,  that  the  rings  are  within  this 
limit.  It  is  not  supposed  that  they  are  the  pulverized 
remains  of  satellites  that  ever  did  actually  exist,  but  rather 
that  the  material  of  which  they  are  composed  is'  subject 
to  such  forces  that  the  mutual  gravitation  of  the  separate 


304      AN   INTRODUCTION  TO  ASTRONOMY    [ch.  ix,  183 

particles  can  never  draw  them  together  into  a  single  body. 
If  they  should  unite  into  a  satellite,  it  would  probably  be 
small,  for  they  are  not  massive  enough  to  have  produced 
by  their  attraction  any  disturbance  of  the  motions  of  the 
satellites  which  can  so  far  be  observed. 

One  more  interesting  thing  remains  to  be  mentioned.  If 
a  meteor  were  to  revolve  in  the  vacant  space  between  the 
rings  known  as  Cassini's  division,  its  period  would  be  nearly 
commensurable  with  the  periods  of  four  of  the  satellites, 
and  would  be  one  half  that  of  Mimas.  Ejrkwood  called 
attention  to  this  relation,  which  is  entirely  analogous  to  that 
found  in  the  case  of  the  planetoids  (Art.  160).  Encke  and 
other  astronomers  have  suspected  that  there  is  a  narrow 
division  between  the  crape  ring  and  the  inner  edge  of  the 
bright  ring,  where  the  period  of  a  revolving  meteor  would 
be  one  third  that  of  Mimas.  More  recently  Lowell  has  been 
convinced  by  his  observations  at  Flagstaff  of  the  existence 
of  several  other  very  narrow  divisions  at  places  where  the 
periods  of  revolving  particles  would  be  simply  commensur- 
able with  the  periods  of  Mimas  or  Enceladus.  But  in  order 
to  secure  perfect  commensurability  he  was  led  to  the  con- 
clusion that  Saturn  is  composed  of  layers  of  different  den- 
sities, and  that  the  inner  ones  are  more  oblate,  and,  there- 
fore, rotate  faster,  than  the  outer  ones. 

184.  On  the  Permanency  of  Saturn's  Rings.  —  The  ques- 
tion at  once  arises  whether  the  meteoric  constitution  of  the 
rings,  in  which  there  is  abundant  opportunity  for  collisions, 
is  a  permanent  one.  The  fact  that  the  rings  exist  and  are 
separated  from  the  planet  by  a  nimiber  of  thousands  of 
miles,  while  beyond  them  there  are  9  satelUtes,  indicates 
that  they  are  not  transitory  in  character.  The  only  cir- 
cumstance that  distinguishes  them  dynamically  from  the 
satellites  is  the  possibility  of  their  colhsions.  If  a  collision 
occurred,  at  least  some  heat  would  be  generated  at  the 
expense  of  their  energy  of  motion.  When  the  revolutionary 
energy  of  a  body  is  decreased,  its  orbit  diminishes  in  size. 


CH.  IX,  185]  THE   PLANETS  305 

Therefore,  when  two  of  the  small  bodies  of  which  Saturn's 
ring  is  composed  coUide,  the  orbit  of  at  least  one  of  them 
must  be  diminished  in  size.  These  colHsions  with  the  accom- 
panjdng  degradation  of  energy  are  probably  taking  place  at 
a  very  slow  rate.  If  so,  the  rings  of  Saturn  are  slowly  shrink- 
ing down  on  the  planet.  It  may  be  that  the  crape  ring  is 
the  result  of  particles  whose  orbits  have  been  reduced  from 
the  larger  dimensions  of  the  bright  ring  by  collisions  with 
other  particles. 

185.  The  Siuf  ace  Markings  and  the  Rotation  of  Saturn.  — 
The  surface  markings  of  Saturn  are  much  hke  those  of 
Jupiter,  though,  of  course,  they  are  not  seen  so  well  because 
of  the  great  distance  of  this  planet.  There  are  a  bright 
equatorial  belt  and  a  number  of  darker  and  broader  belts  in 
the  higher  latitudes,  though  they  are  less  conspicuous  than 
the  belts  on  Jupiter. 

It  has  been  rather  difficult  for  observers  to  find  spots  on 
Saturn  conspicuous  and  lasting  enough  to  enable  them  to 
determine  the  period  of  its  rotation.  From  observations 
made  in  1794  Herschel  concluded  that  its  period  of  rotation 
is  10  hrs.  and  16  m. ;  Hall's  observation  of  a  bright  equatorial 
spot  in  1876  gave  for  this  spot  a  period  of  10  hrs.  and  14  m. 
This  was  generally  adopted  as  the  period  of  Saturn's  rotation, 
particularly  after  it  had  been  verified  by  a  number  of  other 
observers.  But,  in  1903,  Barnard  discovered  some  bright 
spots  in  northern  latitudes,  and  his  observations  of  them, 
together  with  those  of  several  other  astronomers,  showed 
that  these  spots  were  passing  around  Saturn  in  10  hrs.  and 
38  m.  This  difference  in  period  means  that  there  is  a  relative 
drift  between  the  material  of  Saturn's  equatorial  belt  and 
that  of  its  higher  latitudes  of  800  or  900  miles  per  hour. 

In  sharp  contrast  to  the  planet  Jupiter,  the  plane  of  the 
equator  of  Saturn  is  inclined  to  the  plane  of  its  orbit  by  an 
angle  of  27°.  This  is  a  still  higher  inclination  than  those 
found  in  the  case  of  the  earth  and  Mars,  and  would  hardly 
be  expected  in  so  large  a  planet  as  Saturn  after  finding  that 


306      AN   INTRODUCTION  TO  ASTRONOMY    [ch.  ix,  185 

the  axis  of  Jupiter  is  almost  exactly  perpendicular  to  the 
plane  of  its  orbit. 

186.  The  Physical  Condition  and  Seasonal  Changes  of 
Saturn.  —  The  density  of  Saturn  is  about  0.63  on  the  water 
standard.  Consequently,  it  must  be  largely  in  a  gaseous 
condition.  Probably  no  considerable  portion  of  it  is  purely 
gaseous,  for  it  seems  more  likely,  in  view  of  the  fact  that  it 
is  opaque,  that  the  gases  of  which  it  is  composed  are  filled 
with  minute  liquid  particles,  just  as  our  own  atmosphere 
becomes  charged  with  globules  of  water,  forming  clouds. 

The  remarkable  relative  motions  of  the  different  parts  of 
the  surface  of  Saturn  show  that  it  is  at  least  in  a  fluid  state 
and  that  it  is  a  place  of  the  wildest  turmoil.  Doubtless  it  is 
a  world  whose  evolution  has  not. yet  sufficiently  advanced  to 
give  it  any  permanent  markings,  much  less  to  fit  it  as  a  place 
in  any  way  suitable  for  the  abode  of  even  the  lowest  forms  of 
Ufe. 

The  high  inclination  of  the  plane  of  Saturn's  equator  to 
that  of  its  orbit  gives  it  marked  seasonal  changes.  More- 
over, its  orbit  is  rather  more  eccentric  than  the  orbits  of 
the  other  large  planets.  But  it  is  so  far  from  the  sun  that 
it  receives  only  -^  as  much  light  and  heat  per  unit  area  as 
the  earth  receives ;  and  it  follows  that  its  surface  is  very  cold 
vmless  it  has  an  atmosphere  of  remarkable  properties,  or  unless 
a  large  amount  of  heat  is  conveyed  to  it  from  a  hot  interior. 

A  consequence  of  the  rapid  rate  of  rotation  and  low  den- 
sity of  Saturn  is  that  it  is  very  oblate.  The  difference  be- 
tween its  equatorial  and  polar  diameters  is  nearly  6700  miles, 
or  about  10  per  cent  of  its  whole  diameter.  Its  oblateness 
is  so  great  that  it  is  conspicuous  even  through  a  telescope  of 
6  inches'  aperture. 

V.   Uranus  and  Neptune 

187.  The  Satellite  Systems  of  Uranus  and  Neptune.  — 

Uranus  has  four  known  satellites,  two  of  which  were  dis- 
covered by  William  Herschel,  in  1787,  and  the  other  two 


CH.  IX,  189]  THE   PLANETS  307 

by  Lassell,  in  1851.  Their  distances  are  respectively  120,000, 
167,000,  273,000  and  365,000  miles,  and  their  periods  of 
revolution  are  respectively  2.5,  4.1,  8.7,  and  13.5  days. 
Their  diameters  probably  range  between  500  and  1000 
miles.  They  all  move  sensibly  in  the  same  plane,  but  this 
plane  is  inclined  about  98°  to  the  plane  of  the  planet's 
orbit ;  that  is,  if  the  plane  of  the  orbits  of  the  satellites  is 
thought  of  as  having  been  turned  up  from  that  of  the  planet's 
orbit,  the  rotation  has  been  continued  8°  beyond  perpendicu- 
larity, and  the  satellites  revolve  in  the  retrograde  direction. 
Neptune  has  one  known  satellite  which  was  discovered 
by  Lassell,  in  1846.  It  revolves  at  a  distance  of  221,500 
miles  in  a  period  of  5  days  21  hours.  Its  diameter  is  probably 
about  2000  miles.  The  plane  of  its  orbit  is  inclined  about 
145°  to  that  of  the  planet's  orbit;  that  is,  the  inclination 
between  the  two  planes  is  about  35°  and  the  satellite  revolves 
in  the  retrbgrade  direction. 

188.  Atmospheres  and  Albedoes  of  Uranus  and  Neptune. 

—  Very  little  is  known  directly  respecting  the  atmospheres 
of  Uranus  and  Neptune.  Their  low  mean  densities  imply 
that  their  exterior  parts  are  largely  in  the  gaseous  state. 
As  confirmatory  of  -this  conclusion,  the  spectroscope  shows 
that  the  Ught  which  we  receive  from  them  must  have  passed 
through  an  extensive  absorbing  medium  in  addition  to  the 
sun's  atmosphere  and  that  of  the  earth,  through  which  the 
light  from  all  planets  passes.  The  absorbing  effects  of  the 
element  hydrogen  and  water  vapor  are  shown  in  the  spectra 
of  both  planets,  but,  according  to  the  recent  results  of  Slipher, 
more  strongly  in  the  case  of  Neptune  than  in  that  of  Uranus. 
A  number  of  the  other  absorption  bands  are  due  to  unknown 
substances. 

The  albedo  of  Uranus  is  0.63,  and  that  of  Neptune,  0.73. 

189.  The  Periods  of  Rotation  of  Uranus  and  Neptune. 

—  Surface  markings  have  been  seen  on  Uranus  by  Buff  ham. 
Young,  the  Ahdre  brothers,  Perrotin,  Holden,  Keeler,  and 
other  observers,  but  they  have  been  so  indefinite  and  fleeting 


308      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  ix,  189 

that  it  has  not  been  possible  to  draw  any  certain  conclusions 
from  them.  Nevertheless,  so  far  as  they  go,  they  indicate 
that  the  period  of  rotation  of  Uranus  is  10  or  12  hours,  and 
that  the  plane  of  its  equator  is  incUned  som'ething  like  10°  to 
30°  to  the  plane  of  the  orbits  of  the  satellites.  In  1894, 
Barnard  detected  a  sUght  flattening  of  the  disk,  with  the 
equatorial  diameter  inclined  28°  to  the  plane  of  the  orbits 
of  the  satelhtes.  Finally,  in  1912,  V.  M.  Slipher,  at  the 
Lowell  Observatory,  found  by  spectroscopic  means  that 
Uranus  rotates  in  the  direction  of  revolution  of  its  satel- 
lites in  a  period  of  10  hrs.  50  m.  This  result  is  entitled  to 
considerable  confidence. 

No  certain  markings  have  been  seen  on  Neptune,  and, 
consequently,  its  rate  of  rotation  has  not  been  found  by 
direct  means.  But  by  indirect  processes  both  the  position 
of  the  plane  of  its  equator  and  its  rate  of  rotation  have 
been  found,  at  least  approximately.  The  dimensions  and 
mass  of  Neptune  are  known  with  considerable  accuracy. 
Now,  if  the  rate  of  rotation  were  known,  the  equatorial 
bulging  could  be  computed.  Suppose  the  plane  of  the  orbit 
of  the  satellite  were  inclined  to  that  of  the  planet's  equator. 
Then  the  equatorial  bulge  would  perturb  the  motion  of  the 
satellite;  in  particular,  it  would  cause  a  revolution  of  its 
nodes,  and  the  rate  could  be  computed. 

The  problem  of  determining  the  rate  of  rotation  of  Nep- 
tune is  about  the  converse  of  that  which  has  just  been 
described.  The  nodes  of  the  orbit  of  its  satellite  reVolve, 
and  the  manner  of  their  motion  shows  the  existence  of  a 
certain  equatorial  bulge  inclined  about  20°  to  the  plane  of 
the  satellite's  orbit.  The  bulging,  or  ellipticity,  of  the 
planet  is  ^,  indicating,  according  to  the  work  of  Tisserand 
and  Newcomb,  a  rather  slow  rotation  as  compared  to  the 
rates  of  rotation  of  Jupiter  and  Saturn. 

190.  Physical  Condition  of  Uranus  and  Neptune.  —  We 
can  infer  the  physical  conditions  of  Uranus  and  Neptune  only 
from  that  of  other  planets  which  are  more  favorably  situated 


CH.  IX,  190]    .  THE   PLANETS  309 

for  observation.  They  are  probably  in  much  the  same  state 
as  Jupiter  and  Saturn,  though,  possibly,  somewhat  further 
advanced  in  their  evolution  because  of  their  smaller  dimen- 
sions. One  thing  to  be  noticed  is  that  they  receive  rela- 
tively little  light  and  heat  from  the  sun.  The  amounts  per 
unit  area  are  about  ^  and  -^  that  received  by  the  earth. 
If  their  capacity  for  absorbing  and  radiating  heat  were  the 
same  as  that  of  the  earth,  their  temperatures  (Art.  172)  would 
be  respectively  about  —  340°  and  —  364°  Fahrenheit.  Never- 
theless, it  must  not  be  imagined  that  even  Neptune  would 
receive  only  feeble  illumination  from  the  sun.  Although 
the  sun,  as  seen  from  that  vast  distance,  would  subtend  a 
smaller  angle  than  Venus  does  to  us  when  nearest  the  earth, 
the  noonday  illumination  would  be  equal  to  700  times  our 
brightest  moonlight. 

XIII.   QUESTIONS 

1.  Find  by  the  method  of  Art.  172  what  the  mean  temperatures 
of  the  earth  would  be  at  the  distances  of  Mercury  and  Venus. 

2.  If  the  earth  always  presented  the  same  face  toward  the  sun, 
and  if  there  were  no  distribution  of  heat  by  the  atmosphere,  what 
would  be  the  mean  temperature  of  its  illuminated  side?  What 
would  be  the  result  if  the  earth  were  at  the  distance  of  Venus  from 
the  sun? 

3.  If  the  mean  temperature  of  the  equatorial  zone  of  the  earth 
is  85°,  and  if  it  receives,  per  unit  area,  2.5  times  as  much  heat  as  the 
polar  regions,  what  is  the  mean  temperature  of  the  polar  regions, 
neglecting  the  transfer  of  heat  by  the  atmosphere? 

4.  What  would  be  the  mean  temperature  of  the  equatorial 
zone  of  the  earth  at  the  mean  distance  of  Mars  ? 

5.  Suppose  the  mean  temperature  of  the  Thibetan  plateau  at  a 
height  of  15,000  feet  above  sea  level  is  40° ;  what  would  it  be  if 
the  earth  were  at  the  distance  of  Mars  from  the  sun  ? 

6.  Suppose  the  atmosphere  which  a  planet  can  hold  is  propor- 
tional to  its  surface  gravity;  how  does  the  atmosphere  of  Mars 
compare  with  that  of  the  earth  at  an  altitude  of  15,000  feet  above 
sea  level  ? 

7.  Waiving  the  temperature  difficulties  in  the  hypothesis  re- 
garding the  habitability  of  Mars,  what  reasonable  explanation  can 


310      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  ix,  190 

you  give  for  the  fact  that  the  canals  are  always  along  the  arcs  of 
great  circles  ? 

8.  Try  the  experiment  of  Maunder  and  Evans. 

9.  What  would  be  the  total  area  of  400  canals  having  an  aver- 
age width  of  20  miles  and  an  average  length  of  300  miles  ?  Suppose 
to  irrigate  this  area  for  a  season  a  foot  of  water  is  required ;  how 
much  would  this  water  weigh  on  the  earth  ?  On  Mars  ?  Suppose 
a  fall  of  four  feet  per  mile  is  required  to  get  a  flow  in  the  canals  at 
the  necessary  rate ;  suppose  it  is  necessary  to  pump  the  water  out 
of  the  "marshes"  to  a  higher  level  to  get  the  fall;  suppose  the 
pumps  work  10  hours  a  day  for  300  days ;  how  many  horsepower 
of  work  must  they  deliver  ? 


CHAPTER  X 
COMETS   AND   METEORS 

I.   Comets 

191.  General  Appearance  of  Comets.  —  The  planets  are 
characterized  by  the  invariabiHty  of  their  form,  the  sim- 
pUcity  of  their  motions,  and  their  general  similarity  to  one 
another.  In  strong  contrast  to  these  relatively  stable  bodies 
are  the  comets,  whose  bizarre  appearance,  complex  motions, 
and  temporary  visibility  have  led  astronomers  to  devote  to 
them  a  great  amount  of  attention.  Until  the  last  two  cen- 
turies they  were  objects  of  superstitious  terror  which  were 
supposed  to  portend  calamities.  At  least  so  far  as  their 
motions  are  concerned,  they  are  now  known  to  be  as  lawful 
as  the  other  members  of  the  solar  system. 

The  typical  comet  is  composed  of  a  head,  or  coma,  a 
brighter  nucleus  within  the  head  which  is  often  starlike  in 
appearance,  and  a  tail  streaming  out  in  the  direction  oppo- 
site to  the  sun.  The  apparent  size  of  the  head  may  be  any- 
where from  almost  starhke  smallness  to  the  angular  dimen- 
sions of  the  sun.  The  nucleus  is  usually  very  small  and 
bright,  but  the  tail  often  extends  many  degrees  from  the 
head  before  it  gradually  fades  out  into  the  darkness  of  the 
sky.  The  head  is  the  most  distinctive  part  of  the  comet,  for 
it"  is  always  present  and  looks  much  like  a  circular  nebula. 
Either  the  nucleus  or  tail,  or  both,  may  be  absent,  especially 
if  the  comet  is  a  small  one.  Comets  vary  in  brightness  from 
those  which  are  so  faint  that  they  are  barely  visible  through 
large  telescopes  to  those  which  are  so  bright  that  they  may 
be  observed  in  full  daylight,  even  when  almost  in  the  direc- 
tion of  the  sun.    In  spite  of  their  being  sometimes  very 

311 


312       AN  INTRODUCTION  TO  ASTRONOMY    [ch.  x,  191 


Pig.  119.  —  Brooks'  Comet,  Oct.  19,  1911.     Photographed  by  Barnard  at  the 
Yerkes  Observatory. 


CH.  X,  192]  COMETS  AND   METEORS  313 

bright,  they  are  so  nearly  transparent  that  faint  stars  are 
visible  through  them  without  the  slightest  appreciable 
diminution  of  their  light. 

There  are  records  of  about  400  comets  having  been  seen 
before  the  invention  of  the  telescope,  in  1609,  and  more 
than  the  same  number  have  been  observed  since  that  date. 
Astronoihers  now  keep  a  close  watch  of  the  sky,  and  only 
very  faint  ones  can  escape  their  notice.  From  3  to  10  are 
found  yearly.  They  are  lettered  for  each  year  a,  b,  c,  .  .  . 
in  the  order  of  their  discovery,  and  are  numbered  I,  II,  III, 
...  in  the  order  that  they  pass  their  perihelia.  Besides 
this,  they  are  generally  named  after  their  discoverers. 

192.  The  Orbits  of  Comets.  —  In  ancient  times  it  was 
supposed  that  comets  were  malevolent  visitors  prowUng 
through  the  earth's  atmosphere,  bent  on  mischief.  Kepler 
supposed  they  moved  in  straight  lines,  but  Doerfel  showed 
that  the  comet  of  1681  moved  in  a  parabola  around  the  sun 
as  a  focus.  In  1686  Newton  invented  a  graphical  method 
of  computing  comets'  orbits  from  three  or  more  observations 
of  their  apparent  positions.  Better  methods  have  been 
devised  by  Lambert,  Laplace,  Gauss,  and  later  astronomers, 
and  now  there  is  usually  no  difficulty  in  determining  the 
elements  of  an  orbit  from  three  complete  observations  which 
are  separated  bya  few  days. 

The  orbits  of  about  400  comets  have  been  computed, 
and  as  nearly  as  can  be  determined  from  the  imperfect  obser- 
vations on  which  the  computations  of  many  of  them  are 
based,  the  orbits  of  about  300  of  them  are  essentially  para- 
bohc.  In  fact,  they  are  so  generally  parabolic,  or,  at  least, 
extremely  elongated,  that  it  has  been  customary  in  the  pre- 
liminary computations  to  assume  they  are  parabolas.  Of 
the  remaining  cometary  orbits,  nearly  100  have  been  shown 
to  be  distinctly  elliptical  in  shape. 

A  conic  section  is  an  ellipse  if  its  eccentricity  is  less  than 
unity,  a  parabola  if  its  eccentricity  equals  unity,  and  an 
hyperbola  if  its  eccentricity  exceeds  unity.     Since  a  body 


314       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  192 

moving  subject  to  gravitation  may  describe  any  one  of  these 
three  classes  of  orbits,  and  since  the  eccentricity  of  a  parab- 
ola is  the  Umiting  case  between  that  of  an  ellipse  and  that 
of  an  hyperbola,  it  is  infinitely  improbable  that  the  orbit  of 
any  comet  is  exactly  parabolic. 

It  is  important  to  determine  whether  the  eccentricities 
of  the  orbits  of  comets  are  sHghtly  less  than  unity  or  slightly 
greater  than  unity.  In  the  former  case  comets  are  per- 
manent members  of  the  solar  system ;  in  the  latter,  they  are 
only  temporary  visitors.  The  difficulty  in  answering  the 
question  is  not  theoretical,  but  practical.  In  the  first  place, 
comets  are  more  or  less  fuzzy  bodies  and  it  is  difficult  to 
locate  the  exact  positions  of  their  centers  of  gravity.  In 
the  second  place,  they  are  observed  during  only  a  very  small 
part  of  their  whole  periods  while  they  are  in  the  neighbor- 
hood of  the  earth's  orbit.  Generally  they  aretiot  seen  much 
beyond  the  orbit  of  Mars  and  very  rarely  at  the  distance  of 
Jupiter.  For  such  a  small  arc  the  motion  is  sensibly  the 
same  in  a  very  elongated  ellipse,  in  a  parabola,  and  in  an 
hyperbola  whose  eccentricity  is  near  unity,  as  is  evident 
from  Fig.  120. 

More  than  80  comets  move  in  orbits  whose  major  axes 
are  so  short  that  they  will  certainly  return  to  the  sun.  The 
remainder  move  in  exceedingly  elongated  orbits,  and  the 
character  of  their  motion  is  less  certain.  But  it  is  signifi- 
cant that  the  recent  computations  of  Stromgren  show  that 
in  all  cases  in  which  comets  have  been  sufficiently  observed 
to  give  accurate  results  respecting  their  orbits,  they  were 
moving  in  ellipses  when  they  entered  the  solar  system.  At 
the  present  time  there  is  no  known  case  of  a  comet  which 
was  well  observed  for  a  long  time  whose  orbit  was  hyper- 
bolic, and  astronomers  are  becoming  united  in  the  opinion 
that  they  are  permanent  members  of  the  solar  system. 

The  orbits  of  all  the  planets  are  nearly  in  the  same  plane ; 
on  the  other  hand,  the  planes  of  the  orbits  of  the  comets  he 
in  every  possible  direction  and  exhibit  no  tendency  to  paral- 


CH.  X,  192]  COMETS  AND   METEORS  315 

lelism.  The  perihelia  of  the'  orbits  of  comets  are  distributed 
all  around  the  sun,  but  show  a  slight  tendency  to  cluster  in 
the  direction  in  which  the  sun  is  moving  among  the  stars,  a 
fact  which  probably  has  some  connection  with  the  sun's 
motion. 

Some  comets  have  perihelion  points  only  a  few  hundred 
thousand  miles  from  the  surface  of  the  sun,  and  when  nearest 


or= 


S» 


^ 


Fig.  120.  —  Similarity  of  elongated  ellipses,  parabolas,  and  hyperbolas  in 
the  vicinity  of  the  orbit  of  the  earth. 

the  sun  they  actually  pass  through  its  corona  (Art.  238). 
About  25  comets  pass  within  the  orbit  of  Mercury ;  nearly 
three  fourths  of  those  which  have  been  observed  come 
•within  the  orbit  of  the  earth ;  very  few  so  far  seen  are  per- 
manently without  the  orbit  of  Mars,  and  all  known  comets 


316       AN   INTRODUCTION   TO  ASTRONOMY    [ch.  x.  192 

come  within  the  orbit  of  Jupiter.  This  does  not  mean  that 
there  are  no  comets  with  great  perihehon  distances,  or  even 
that  those  with  perihehon  distances  greater  than  the  distance 
from  the  earth  to  the  sun  are  not  very  numerous.  Comets 
are  relatively  inconspicuous  objects  until  they  come  con- 
siderably within  the  orbit  of  Mars.  Sometimes  their  bright- 
ness increases  a  hundred  thousandfold  while  they  move  from 
the  orbit  of  Mars  to  that  of  Mercury.  Consequently,  even 
if  comets  whose  perihelia  are  beyond  the  orbit  of  Mars  were 
very  numerous,  not  many  of  them  would  be  observed. 

193.  The  Dimensions  of  Comets.  —  After  the  orbits  of 
comets  have  been  computed  so  that  their  distances  from  the 
earth  are  known,  their  actual  dimensions  can  be  determined 
from  their  apparent  dimensions.  It  has  been  found  that 
the  head  of  a  comet  may  have  any  diameter  from  10,000 
miles  up  to  more  than  1,000,000  miles.  The  most  remark- 
able thing  about  the  liead  of  a  comet  is  that  it  nearly  always 
contracts  as  the  comet  approaches  the  sun,  and  expands 
again  when  the  comet  recedes.  The  variation  in  volume  is 
very  great,  the  ratio  of  the  largest  to  the  smallest  sometimes 
being  as  great  as  100,000  to  1.  John  Herschel  suggested 
that  the  contraction  may  be  only  apparent,  the  outer  layers 
of  the  comet  becoming  transparent  as  it  approaches  the  sun. 
This  suggestion  contradicts  the  appearances  and  seems  to  be 
extremely  improbable. 

The  nucleus  of  a  comet  may  be  so  small  as  to  be  scarcely 
visiblej  say  100  miles  in  diameter,  or  it  may  be  as  large  as 
the  earth.  For  example,  William  Herschel  observed  the 
great  comet  of  1811  when  its  head  was  more  than  500,000 
miles  in  diameter,  while  its  nucleus  measured  only  428  miles 
across.  The  nuclei  vary  in  size  during  the  motion  of  comets, 
but  the  change  is  quite  irregular  and  no  law  of  variation  has 
been  discovered. 

The  tails  of  comets  are  inconceivably  large.  Their  diam- 
eters are  counted  by  thousands  and  tens  of  thousands  of 
miles  where  they  leave  the  heads  of  comets,  and  by  tens  of 


CH.  X,  194] 


COMETS  AND   METEORS 


317 


thousands  or  hundreds  of  thousands  of  miles  in  their  more 
remote  parts.  They  vary  in  length  from  a  few  million 
miles,  or  even  less,  up  to  more  than  a  hundred  million  of 
miles.  In  volume,  the  tails  of  comets  are  thousands  of 
times  greater  than  the  sun  and  all  the  planets  together. 
The  strangest  thing  about  them  is  that  they  point  almost 
directly  away  from  the  sun  whichever  way  the  comet  may 
be  going.  That  is,  when  the  comet  is  approaching  the  sun, 
the  tails  trail  behind  like  the  smoke 
from  a  locomotive;  when  the  comet 
is  receding,  they  project  ahead  like 
the  rays  from  the  head  light  on  a 
misty  night.  When  a  comet  is  far 
from  the  sun,  its  tail  is  small,  or  may 
be  entirely  absent;  as  it  approaches 
the  sun,  the  tail  develops  in  dimensions 
and  splendor,  and  then  diminishes 
again  on  its  recession  from  the  sun. 

194.  The  Masses  of  Comets.  — 
Comets  give  visible  evidence  of  re- 
markable tenuity,  but  their  volumes 
are  so  great  that,  if  their  densities 
were  one  ten-thousandth  -of  that  of 
air  at  the  surface  of  the  earth,  their  masses  in  many  cases 
would  be  comparable  to  the  masses  of  the  planets. 

The  masses  of  comets  are  determined  from  their  attrac- 
tions for  other  bodies  (Arts.  19,  154).  Or,  rather,  their  lack 
of  appreciable  mass  is  shown  by  the  fact  that  they  do  not 
produce  observable  disturbing  effects  in  the  motions  of 
bodies  near  which  they  pass.  Many  comets  have  had  their 
orbits  entirely  changed  by  planets  without  producing  any 
sensible  effects  in  return.  Since,  according  to  the  third  law 
of  motion,  action  between  two  bodies  is  equal  and  opposite, 
it  follows  that  the  masses  of  comets  are  very  small,  probably 
not  exceeding  one  millionth  that  of  the  earth. 

One  of  the  most  striking  examples  of  the  feeble  gravita- 


FiG.  121.  —  The  tails  of 
comets  are  always  di- 
rected away  from  the 
sun. 


318        AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  194 

tional  power  of  comets  was  furnished  by  the  one  discovered 
by  Brooks  in  1889.  It  had  passed  through  Jupiter's  satelhte 
system  in  1886  without  interfering  sensibly  with  the  motions 
of  these  bodies,  although  its  own  orbit  was  so  transformed 
that  its  period  was  reduced  from  27  years  to  7  years. 

195.  Families  of  Comets.  —  Notwithstanding  the  great 
diversities  in  the  orbits  of  comets,  there  are  a  few  groups 
whose  members  seem  to  have  some  intimate  relation  to  one 
another,  or  to  the  planets.  There  are  two  types  of  these 
groups,  and  they  are  known  as  comet  families. 

Families  of  the  first  type  are  made  up  of  comets  which 
pursue  nearly  identical  paths.  The  most  celebrated  family 
of  this  type  is  composed  of  the  great  comets  of  1668,  1843, 
1880,  and  1882.  A  much  smaller  one  seen  in  1887  probably 
should  be  added  to  this  list.  Their  orbits  were  not  only 
nearly  identical,  but  the  comets  themselves  were  very  simi- 
lar in  every  respect.  They  came  to  the  sun  from  the  direc- 
tion of  Sirius  —  that  is,  from  the  direction  away  from  which 
the  sun  is  moving  with  respect  to  the  stars  —  and  escaped 
the  notice  of  observers  in  the  northern  hemisphere  until 
they  were  near  perihehon.  They  passed  half  way  around 
the  sun  in  a  few  hours  at  a  distance  of  less  than  200,000 
miles  from  its  surface,  moving  at  the  enormous  velocity  of 
more  than  350  miles  per  second.  Their  tails  extended  out 
in  dazzling  splendor  100,000,000  miles  from  their  heads. 

One  might  think  that  the  various  members  of  a  comet 
family  are  but  the  successive  appearances  of  the  same  comet ; 
but  such  is  not  the  case,  for  the  observations  show  that 
though  their  orbits  may  be  ellipses,  their  periods  are  at 
least  600  or  800  years.  This  means  that  they  recede  to 
something  like  five  times  the  distance  of  Neptune  from  the 
sun.  The  most  plausible  theory  seems  to  be  that  they  are 
the  separate  parts  of  a  great  comet  which  at  an  earlier  visit 
to  the  sun  was  broken  up  by  tidal  disturbances. 

Families  of  the  gecond  type  are  made  up  of  comets  whose 
orbits  have  their  aphelion  points  and  the  ascending  and 


CH.  X,  195] 


COMETS  AND  METEORS 


319 


descending  nodes  of  their  orbits  near  the  orbits  of  the  planets. 
About  30  comets  have  their  aphelia  near  Jupiter's  orbit, 
and  are  known  as  Jupiter's  family  of  comets,  Fig.  122. 
Their  orbits  are,  of  course,  all  elliptic,  and  their  periods  are 
from  3  to  8  years.  They  move  around  the  sun  in  the  same 
direction  that  the  planets  revolve.    Half  of  them  have  been 


^y-^-Jtt^v"        y 


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Tim  V  (B»' 

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> 

Fig.  122.  —  Jupiter's  family  of  comets  (Popular  Astronomy). 

seen  at  two  or  more  perihelion  passages.  These  comets  are 
all  inconspicuous  objects  and  entirely  invisible  to  us  except 
when  they  are  near  the  earth. 

Saturn  has  a  family  of  2  comets,  Uranus  a  family  of  3, 
and  Neptune  a  family  of  6  members.  The  terrestrial  planets 
do  not  possess  comet  families.  There  are;  according  to  the 
statistical  study  of  W.  H.  Pickering,  two  or  three  groups  of 


320        AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  195 

comets  whose  aphelia  are  several  times  the  distance  of  Nep- 
tune from  the  sun,  suggesting,  possibly,  the  existence  of 
planets  at  these  respective  distances. 

196.  The  Capture  of  Comets.  —  A  very  great  majority 
of  comets  move  in  sensibly  parabolic  orbits  whose  positions 
have  no  special  relations  to  the  positions  of  the  orbits  of  the 
planets.  But  the  orbits  of  nearly  all  those  comets  which 
are  elKptical  and  not  exceedingly  elongated  he  near  the 
plane  of  the  planetary  orbits  and  have  their  apheha  near 
the  orbits  of  the  planets.  These  facts  suggest  that  the 
orbits  of  comets  moving  in  these  ellipses  have  been  changed 
from  parabolas  or  very  elongated  elKpses  by  the  disturbing 
action  of  the  planet  near  whose  orbit  their  aphehon  points 
lie.  This  question  of  the  transformation  of  orbits  of  comets 
was  first  discussed  by  Laplace,  who  found  that  if  a  comet 
which  is  approaching  the  sun  on  a  parabolic  or  elongated 
elliptical  orbit  passes  closely  in  front  of  a  planet,  its  motion 
will  be  retarded  so  that  it  will  subsequently  move  in  a 
shortened  elhptical  orbit,  at  least  until  it  is  disturbed 
again. 

Suppose  a  comet  approaches  the  sun  in  a  sensibly  para- 
bolic orbit  and  passes  closely  in  front  of  a  planet  so  that  its 
orbit  is  reduced  to  an  ellipse.  It  is  then  said  to  have  been 
captured.  It  will  in  the  course  of  time  pass  near  the  planet 
again,  when  its  orbit  may  be  still  further  reduced ;  or,  its 
orbit  may  be  elongated  and  it  may  possibly  be  driven  from 
the  solar  system  on  a  parabola  or  an  hyperbola. 

It  is  a  generally  accepted  theory  that  the  members  of  the 
comet  families  of  the  various  planets  have  been  captured 
by  the  method  described.  Jupiter  has  a  larger  family  of 
comets  than  any  other  planet  because  of  its  greater  mass 
and  also  because,  if  a  comet  were  captured  originally  by  any 
planet  beyond  the  orbit  of  Jupiter,  it  would  yet  be  possible 
for  Jupiter  to  reduce  its  orbit  still  further.  On  the  other 
hand,  when  Jupiter  has  captured  a  comet  and  made  it  a 
member  of  its  own  family,  it  is  far  within  the  orbit  of  the 


CH.  X,  196]  COMETS  AND  METEORS  321 

remoter  planets  and  is  no  longer  subject  to  capture  by  them. 
The  planets  beyond  the  orbit  of  Jupiter  have  a  few  comets 
each,  and  the  clustering  of  the  aphelia  of  comets  at  still 
more  remote  distances  has  suggested  the  existence  of  planets 
as  yet  undiscovered  (Art.  161).  The  terrestrial  planets  have 
no  comet  families  partly  because  their  masses  are  small  com- 
pared to  that  of  the  sun,  and  partly  because  comets  cross 
their  orbits  at  very  great  speed. 

The  masses  of  the  planets  are  not  great  enough  to  reduce 
a  parabolic  comet  to  membership  in  their  own  families  at 
one  disturbance.  The  matter  is  illustrated  by  Brooks'  comet, 
1889-V,  whose  period,  according  to  the  computations  of 
Chandler,  was  reduced  by  Jupiter,  in  1886,  from  27  years 
to  7  years.  Lexell's  comet,  of  1770,  furnishes  an  example 
of  a  disturbance  of  the  opposite  character.  In  1770  it  was 
moving  in  an  elliptical  orbit  with  a  period  of  5.5  years ;  but 
in  1779  it  approacjied  near  to  Jupiter,  its  orbit  was  enlarged, 
and  it  has  never  been  seen  again. 

When  a  planet  captures  a  comet,  the  former  reduces  the 
dimensions  of  the  orbit  of  the  latter,  but  the  latter  still  re- 
volves around  the  sun.  The  question  arises  whether  a  planet 
might  not  capture  a  comet  in  a  more  fundamental  sense ; 
that  is,  reduce  its  orbit  so  that  it  would  become  a  satelhte  of 
the  planet.  It  has  been  repeatedly  suggested  that  the 
planets  may  have  captured  their  satellites  in  this  manner. 
The  answer  to  this  suggestion  is  that  a  planet  cannot  capture 
a  comet  and  make  it  into  a  satellite  simply  by  its  own  grav- 
tation  and  that  of  the  sun.  The  only  possibility  is  that  the 
comet  should  encounter  resistance  in  a  very  special  manner, 
and  even  then  the  ptoblem  presents  serious  difficulties.  No 
small  resistance  would  be  sufficient  because  the  motion  of  a 
comet  around  the  sun  in  a  parabolic  orbit  is  much  greater 
than  it  would  be  in  a  satellite  orbit ;  and,  in  order  that  resist- 
ance should  reduce  the  velocity  by  the  required  amount,  it 
would  be  necessary  for  the  comet  to  encounter  so  much 
material  that  its  mass  would  grow  several  fold. 


322       AN  INTRODUCTION  TO  ASTRONOMY    [ch.  x,  197 

197.  On  the  Origin  of  Comets.  —  The  similarities  of  the 
motions  of  the  various  planets  point  to  the  conclusion  that 
they  had  a  common  origin,  and  the  agreement  of  the  direc- 
tion of  the  rotation  of  the  sun  with  their  direction  of  revolu- 
tion indicates  that  they  have  been  associated  with  the  sun 
throughout  their  whole  evolution.  This  line  of  reasoning 
does  not  lead  to  the  inference  that  the  comets  belong  to  the 
planetary  family.  They  may  have  had  quite  a  different 
origin ;  at  any  rate,  most  of  them  recede  from  the  sun  to 
regions  several  times  as  remote  as  the  planet  Neptune. 

It  was  formerly  supposed  that  comets  are  merely  small, 
wandering  masses  which  pass  from  star  to  star,  visiting  our 
sun  but  once.  The  intervals  of  time  required  for  such  excur- 
sions are  enormously  greater  than  has  generally  been  sup- 
posed. For  example,  the  great  comet  of  1882  came  almost 
exactly  from  the  direction  of  Sirius  and  returned  again  in 
the  same  direction.  Suppose  the  comet  moved  under  the 
attraction  of  Sirius  until  it  had  passed  over  half  of  the  dis- 
tance from  Sirius  to  the  sun,  and  that  it  then  moved  sen- 
sibly under  the  attraction  of  the  sun.  Although  Sirius  is 
one  of  the  nearest  known  stars  in  all  the  sky,  it  is  found  that 
it  would  take  70,000,000  years  to  describe  this  part  of  its 
orbit.  About  twice  this  period  of  time  would  be  required 
for  it  to  come  from  Sirius  to  the  sun,  and  eight  times  this 
immense  interval  for  a  comet  to  come  from  a  star  four  times 
as  far  away.  These  figures  do  not  disprove  the  theory  that 
comets  wander  from  star  to  star,  but  they  show  that  if  this 
hypothesis  is  true,  then  comets  spend  most  of  their  time  in 
traveling  and  but  little  in  visiting. 

If  the  comets  moved  from  star  to  star,  their  orbits  with 
respect  to  the  sun  would  never  be  elliptical  until  after  they 
had  been  captured ;  they  would,  indeed,  nearly  always  be 
strongly  hyperbohc  because  the  stars  are  moving  with  respect 
to  one  another  with  velocities  which  correspond  to  hyper- 
bolic speed  for  comets  at  such  great  distances.  The  fact 
that  no  comet  out  of  the  hundreds  whose  orbits  have  been 


CH.  X,  198]  COMETS  AND   METEORS  323 

computed  has  moved  in  a  sensibly  hyperbolic  orbit  points 
strongly  to  the  conclusion  that  comets  have  been  permanent 
members  of  the  solar  system.  They  are  possibly  the  remains 
of  the  far  outlying  masses  of  a  nebula  from  which  the  solar 
system  may  have  been  developed.  With  increasing  proof 
that  they  are  actually  permanent  members  of  the  solar  sys- 
tem, their  importance  in  connection  with  the  question  of  its 
origin  and  evolution  continually  increases. 

198.   Theories  of  Comets'  Tails.  —  The  fact  that  the  tails 
of  comets  usually  project  almost  directly  away  from  the 
sun  indicates  that  they  are  in 
some   way   acted    upon   by    a  _,„-'--'---- 

repelling  force  emanating  from  _,/-V'",. 

the  sun.     The  intensity  of  this      tosun       o^r"'' "' 

repulsion  has  been  computed  in  ' v'-'^,- 1  ^  ^ 

a  number  of  cases  by  Barnard  ''''C>  "~-... 

and  others  from  the  accelera-  '"^'^V," 

tions  which  masses  have  under-  '--".v.' 

gone  which  were  receding  from  Fig-  123.  — The  repulsion  theory 

°  01  the  origin  of  comets  tails. 

the  heads  of  comets  along  their 

tails.  These  accelerations  have  been  determined  by  com- 
paring photographs  of  the  comets  taken  at  different  times 
separated  by  short  intervals. 

It  was  suggested  by  Olbers  as  early  as  1812  that  the  repul- 
sive force  which  apparently  produces  the  tails  of  comets  may 
be  electrical  in  character.  This  theory  has  been  taken  up 
and  systematically  developed  by  Bredichin,  of  Moscow. 
According  to  it,  the  sun  and  comet  nuclei  both  repel  the 
material  of  which  the  tails  of  comets  are  composed.  Those 
particles  which  leave  the  nuclei  in  the  direction  away  from 
the  sun  continue  on  in  straight  lines ;  those  which  leave  in 
other  directions  are  gradually  bent  back  by  the  force  from 
the  sun  and  form  the  outer  parts  of  the  tails,  as  shown  in 
Fig.  123.  The  resulting  tails,  especially  if  they  are  very 
long,  are  slightly  curved  because  the  motion  of  the  comet 
is  somewhat  athwart  the  line  along  which  the  repelled  par- 


324       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  198 

tides  move,  that  is,  the  Une  from  the  sun  through  the  nucleus 
(see  Fig.  121). 

Electrical  repulsion  acts  on  the  surfaces  of  particles,  while 
gravitation  depends  on  their  masses.  Therefore,  while  large 
masses  are  attracted  by  the  sun  more  than  they  are  elec- 
trically repelled,  the  opposite  may  be  true  for  small  particles, 
and  the  electrical  repulsion  is  relatively  stronger  the  smaller 
they  are.  Consequently,  the  tails  which  are  produced  out 
of  small  particles  will  be  more  nearly  straight  than  those 
which  are  composed  of  larger  particles.  Bredichin  advanced 
the  theory  that  the  long,  straight  tails  are  due  to  hydrogen 
gas,  the  ordinary  slightly  curved  tails  to  hydrocarbon  gases, 
and  the  short,  stubby,  and  much  curved  tails  to  vapors  of 
metals.  Spectroscopic  observations  have  to  a  considerable 
extent  confirmed  these  conclusions.  Some  comets  have  tails 
of  more  than  one  type,  as  for  example  Delavan's  comet 
(Fig.  124). 

If  the  electrical  repulsion  theory  is  adopted,  the  question 
at  once  arises  why  the  sun  and  the  materials  of  which  the 
tails  of  comets  are  composed  are  similarly  electrified.  A 
plausible  answer  to  this  question  can  be  given.  At  least" 
the  hydrogen  in  the  sun's  atmosphere  seems  to  be  negatively 
electrified.  Suppose  a  comet  approaches  the  sun  from  a 
remote  part  of  space  without  an  electrical  charge.  Labora- 
tory experiments  show  that  the  ultra-violet  rays  from  the 
sun,  striking  on  the  nucleus  of  the  comet,  will  probably  drive 
off  negatively  charged  particles  which  will  be  repelled  by 
the  negative  charge  of  the  sun,,  and  they  will  thus  form  a 
tail  for  the  comet.  The  repulsion  will  depend  upon  the 
size  of  the  particles  and  the  electrical  potential  of  the  sun. 
After  the  negatively  electrified  particles  have  been  driven 
off,  the  nucleus  will  be  positively  charged  and,  consequently, 
will  be  electrically  attracted  by  the  sun.  But  since  the  par- 
ticles driven  off  will  be  only  an  exceedingly  small  part  of  the 
whole  comet,  this  attraction  will  not  be  great  enough  sen- 
sibly to  alter  the  comet's  motion. 


CH.  X,  198] 


COMETS  AND  METEORS 


325 


Fig.  124.- 


-Delavan's  comet,  Sept.  28,  1914,  showing  a  long,  straight  taU 
and  one  having  considerable  curvature  (Barnard). 


326       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  198 

Another  theory  which  merits  careful  attention  is  that  the 
particles  which  constitute  comets'  tails  are  driven  off  by  the 
pressure  of  the  sun's  hght.  According  to  Clerk-Maxwell's 
electromagnetic  theory,  hght  exerts  a  pressure  upon  bodies 
upon  which  it  falls  which  is  proportional  to  the  hght  energy 
in  a  unit  of  space.  For  bodies  of  considerable  magnitude 
the  pressure  is  relatively  very  small,  though  it  has  been 
detected  by  Nichols  and  Hull ;  ^  but  for  minute  bodies,  say  a 
ten-thousandth  of  an  inch  in  diameter,  the  light  pressure 
may  greatly  exceed  the  sun's  attraction.  For  still  smaller 
bodies  the  light  pressure  becomes  relatively  larger  until  their 
diameters  are  approximately  equal  to  a  wave  length  of  light, 
say,  one  fifty-thousandth  of  an  inch.  Then,  as  Schwarzschild 
has  shown,  the  light  pressure  decreases  relatively  to  the 
force  of  gravitation.  Consequently,  if  the  particles  are  very 
small  the  attraction  will  more  than  equal  the  repulsion. 

But  it  has  been  shown  more  recently  by  Lebedew  that 
there  is  light  pressure  upon  gases,  in  which  the  diameters 
of  the  molecules  are  always  a  very  small  fraction  of  a 
wave  length  of  light,  and  that  the  pressure  is  proportional  to 
the  amount  of  energy  which  the  gas  absorbs.  Consequently, 
it  is  not  necessary  to  assume  that  the  particles  of  which  the 
tails  of  comets  are  composed  are  larger  than  molecules. 

It  is  generally  supposed  by  astronomers  that  both  elec- 
trical repulsion  and  light  pressure  are  factors  in  the  produc- 
tion of  comets'  tails.  Nevertheless,  there  are  outstanding 
phenomena  which  these  theories  do  not  explain.  In  the 
first  place,  there  is  no  adequate  explanation  of  the  luminosity 
of  comets'  tails.  As  comets  approach  the  sun,  their  tails 
increase  in  brightness  much  more  rapidly  than  they  should 
if  they  were  shining  only  by  reflected  light.  The  luminosity 
of  such  exceedingly  tenuous  bodies  whose  density  is  doubt- 
less far  less  than  that  in  the  best  vacuum  tubes  of  the  present 
time  can  scarcely  be  explained  as  a  temperature  effect. 
And  still  more  embarrassing  to  these  theories  are  the  facts 
that  comets'  tails  do  not  always  point  directly  away  from 


CH.  X,  199]  COMETS  AND  METEORS  327 

the  sun,  and  that  sometimes  they  change  their  direction  by 
a  number  of  degrees  in  a  very  short  time.  For  example, 
Barnard  took  photographs  of  Brooks's  comet,  1893-IV, 
on  November  2  and  November  3.  In  this  interval  the  comet 
moved  forward  in  its  orbit  about  1°;  and,  consequently, 
according  to  these  theories,  the  direction  of  its  tail  should 
have  changed  about  1°.  But  there  was  an  actual  change  of 
direction  of  the  tail  of  16°  which  has  not  been  explained. 
There  are  also  sudden  and  great  changes  in  the  character 
and  luminosity  of  comets'  tails  which  no  theory  explains. 
Sometimes  secondary  tails  are  developed  with  great  rapidity, 
making  an  angle  of  as  much  as  45°  with  the  line  joining  the 
comet  with  the  sun.  Obviously  much  remains  to  be  learned 
in  connection  with  the  tails  of  comets. 

199.  The  Disintegration  of  Comets.  —  The  particles  that 
leave  the  head  of  a  comet  to  form  its  tail  never  unite  with 
it  again.  In  this  way,  at  each  reappearance  of  a  comet,  that 
part  of  the  material  which  goes  to  form  its  tail  is  dispersed 
into  space ;  and,  as  the  quantity  remaining  becomes  reduced, 
the  comet  becomes  less  and  less  conspicuous.  Possibly  this 
is  one  of  the  reasons  why  Halley's  comet  in  1910  was  not 
such  a  remarkable  object  as  it  seems  to  have  been  in  some 
of  its  earlier  apparitions. 

There  is  another  way  in  which  comets  disintegrate.  Since 
their  masses  are  very  small,  the  mutual  attractions  of  their 
parts  are  not  sufficient  to  hold  them  together  if  they  are 
subject  to  strong  disturbing  forces.  When  they  pass  near 
the  sun,  they  are  elongated  by  enormous  tides.  In  fact,  if 
they  pass  within  Roche's  hmit  (Art.  183),  the  tidal  forces  ex- 
ceed their  self  gravitation  unless  they  are  as  dense  as  the  sun. 
Comets  have  such  exceedingly  low  density  that  the  Kmits  of 
tidal  disintegration  for  them  must  be  very  great.  Conse- 
quently, when  a  comet  passes  near  the  sun,  the  tidal  forces 
to  which  it  is  subject  tend  to  tear  it  into  fragments,  which, 
of  course,  may  be  assembled  again  by  their  mutual  gravita- 
tion after  they  have  receded  far  from  the  sun.    But  on 


328      AN  INTRODUCTION   TO  ASTRONOMY    [ch.  x,  199 

their  way  out  they  may  pass  near  a  planet  which  will  exert 
analogous  forces,  and  may  so  disorganize  them  that  they 
will  never  again  be  united  into  a  single  body. 

The  theory  which  has  just  been  outlined  is  clear.  Now 
what  have  been  the  observed  facts?  Biela's  comet  was 
broken  into  two  parts  by  some  unknown  forces,  and  the  two 
components  subsequently  traveled  in  independent  paths. 
The  great  comet  of  1882  was  seen  to  have  a  number  of  out- 
lying fragments  when  it  was  in  the  vicinity  of  the  sun,  and 
many  other  comets  have  exhibited  analogous  phenomena. 

Another  source  of  disturbance  to  which  comets  are  sub- 
ject is  the  scattered  meteoric  material  which  may  more  or 
less  fill  the  space  among  the  planets.  The  phenomenon  of 
the  zodiacal  light  gives  an  almost  certain  proof  of  its  exten- 
sive existence.  Such  scattered  particles  would  have  little 
effect  on  a  dense  body  like  a  planet,  but  might  cause  serious 
disturbances  in  a  tenuous  comet.  In  fact,  there  are  many 
instances  in  which  comets  and  comets'  tails  seem  to  have 
been  subjected  to  unknown  exterior  forces.  They  are  now 
and  then  more  or  less  broken  up,  and  occasionally  the  tails 
of  comets  have  been  apparently  cut  off  and  brushed  aside. 

Many  comets  which  have  been  observed  at  two  or  three 
perihelion  passages  have  been  found  to  be  fainter  at  each 
successive  return  than  they  were  at  the  preceding,  and  some 
have  eventually  entirely  disappeared.  It  seems  to  be  a  safe 
conclusion  that  comets  are  slowly  disintegrated  under  the 
disturbing  forces  of  the  sun  and  planets  and  the  resisting 
meteoric  material  which  they  may  encounter.  As  confirma- 
tory of  this  view,  it  may  be  noted  that  the  members  of  Jupi- 
ter's family  have  small  tails  or  none  at  all ;  that  this  comet 
family  does  not  contain  as  many  members  as  might  be  ex- 
pected ;  and  that  a  number  of  comets  have  totally  disap- 
peared, presumably  by  disintegration. 

200.  Historical  Comets.  —  In  this  article  some  of  those 
comets  will  be  briefly  described  which  have  exhibited  phe- 
nomena  of  unusual   interest.     The   enumeration   of  their 


CH.  X,  200] 


COMETS  AND  METEORS 


329 


peculiarities  will  illustrate  the  general  statements  which  have 
preceded,  and  will  give  additional  information  respecting 
these  remarkable  objects. 

The  Comet  of  1680.  —  The  comet  of  1680  was  the  first  one 
whose  orbit  was  computed  on  the  basis  of  the  law  of  gravi- 
tation. Newton  made  the  calculations  and  found  that  its 
period  of  revolution  was  about  600  years.  It  is  one  of  the 
family  of  comets  mentioned  in  Art.  195.  At  its  periheHon 
it  passed  through  the  sun's  corona  at  a  distance  of  only 
140,000  miles  from  its  surface.  It  flew  along  this  part  of  its 
orbit  at  the  rate  of  370 
miles  per  second,  and 
its  tail,  100,000,000  miles 
long,  changed  its  direc- 
tion to  correspond  with 
the  motion  of  the  comet 
in  its  orbit. 

The  Great  Comet  of 
1811. — The  great  comet 
of  1811  was  visible  from 
March  26,  1811,  until 
August  17, 1812,  and  was 
carefully  observed  by  William  Herschel.  He  discovered  from 
the  changes  in  its  brightness,  that  it  shone  partly  by  its  own 
light;  for  its  brilUance  increased  as  it  approached  the  sun 
more  rapidly  than  it  would  have  done  if  it  had  been  shining 
entirely  by  reflected  Ught.  At  one  time  its  tail  was 
100,000,000  miles  long  and  15,000,000  miles  in  diameter. 
The  phenomena  connected  with  it  suggested  to  Olbers  the 
electrical  repulsion  theory  of  comets'  tails. 

Encke's  Comet  (1819).  —  Encke's  comet  was  the  first 
member  of  Jupiter's  family  to  be  discovered,  and  it  has  a 
shorter  period  (3.3  years)  than  any  other  known  comet. 
At  its  brightest  it  was  an  inconspicuous  telescopic  object 
(Fig.  125),  but  it  is  noted  for  the  fact  that  its  period  was 
shortened,   presumably   by   encountering  some  resistance. 


Fig.   125.  —  Encke's  comet  (Barnard). 


330      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  x,  200 

about  2.5  hours  at  each  revolution  until  1868;  since  that 
time  the  change  in  the  period  of  revolution  has  been  only- 
one  half  as  great.  The  change  in  volume  of  Encke's  comet 
at  times  was  extraordinary.  On  October  28,  1828,  it  was 
135,000,000  miles  from  the  sun  and  had  a  diameter  of  312,000 
miles;  on  December  24,  its  distance  was  50,000,000  miles 
from  the  sun,  and  its  diameter  was  only  14,000  miles ;  while 
at  its  perihehon  passage,  on  December  17,  1838,  at  a  dis- 
tance of  32,000,000  miles,  its  diameter  was  only  3000  miles. 
That  is,  at  one  time  its  volume  was  more  than  a  milhon 
times  greater  than  it  was  at  another. 

Biela's  Comet  (1826).  —  Biela's  comet  is  also  a  srnall 
member  of  Jupiter's  family  and  has  a  period  of  about  6.6 
years.  At  its  appearance  in  1846,  it  presented  no  unusual 
phenomena  until  about  the  20th  of  December,  when  it  was 
considerably  elongated.  By  the  first  of  January  it  had  sepa- 
rated into  two  distinct  parts  which  traveled  along  in  parallel 
orbits  at  a  distance  of  about  160,000  miles  from  each  other. 
At  this  time  the  two  parts  were  undergoing  considerable 
changes  in  brightness,  usually  alternately,  and  sometimes 
they  were  connected  by  a  faint  stream  of  hght.  At  their 
appearance  in  1852  the  two  components  were  1,500,000  miles 
apart,  and  they  have  never  been  seen  again,  although  searched 
for  very  carefully.  De  Vico's  comet,  of  1844,  and  Brorsen's 
comet,  of  1846,  are  also  comets  which  have  disappeared, 
the  former  having  been  observed  but  once,  and  the  latter 
but  four  times  after  its  discovery. 

Donati's  Comet  (1858).  —  Donati's  comet  was  one  of  the 
greatest  comets  of  the  nineteenth  century.  It  was  visible 
with  the  unaided  eye  for  112  days,  and  through  a  telescope 
for  more  than  9  months.  Its  tail,  which  was  more  than 
54,000,000  miles  long,  at  one  time  subtended  an  angle  of  more 
than  30°  as  seen  from  the  earth.  It  moved  in  the  retro- 
grade direction  in  an  orbit  with  a  period  of  more  than  2000 
years,  and  at  its  aphelion  its  distance  from  the  sun  was 
more  than  5.3~times  that  of  Neptune. 


CH.  X,  200]  COMETS  AND  METEORS  331 

Tebbutt's  Comet  (1861).  —  Tebbutt's  comet  was  of  great 
dimensions,  but  is  noteworthy  chiefly  because  the  earth 
passed  through  its  tail.  As  could  have  been  anticipated 
from  the  excessive  tenuity  of  comets'  tails,  the  earth  experi- 
enced no  sensible  effects  from  the  encounter.  The  earth 
must  have  passed  through  the  tails  of  comets  many  times  in 
geological  history,  and  there  is  no  evidence  whatever  that  it 
has  ever  been  disturbed  by  them.  In  fact,  if  a  comet  should 
strike  the  earth,  head  on,  it  is  probable  that  the  result  would 
not  be  disastrous  to  the  earth. 

The  Great  Comets  of  1880  and  1882.  —  The  comets  of  1880 
and  1882  were  two  splendid  members  of  the  most  remark- 
able known  family  of  comets  which  travel  in  the  same  orbit. 
Both  of  these  comets,  as  well  as  the  earlier  members  of  the 
same  family,  are  noteworthy  for  their  vast  dimensions,  their 
great  brilhancy,  and  their  close  approach  to  the  sun.  The 
comet  of  1882  was  observed  both  before  and  after  peri- 
helion passage.  Although  it  swept  through  several  hundred 
thousand  miles  of  the  sun's  corona,  its  orbit  was  not  sensibly 
altered.  Yet  it  gave  evidence  of  having  been  subject  to 
violent  disrupting  forces.  After  perihelion  passage  it  was 
observed  to  have  as  many  as  5  nuclei,  while  Barnard  and 
other  observers  saw  in  the  immediate  vicinity  as  many  as 
6  or  8  small  comet-like  masses,  apparently  broken  from  the 
large  body,  traveling  in  orbits  parallel  to  it. 

Morehouse's  Comet  (1908).  —  On  September  1,  1908, 
Morehouse,  at  the  Yerkes  Observatory,  discovered  the  third 
comet  of  the  year.  It  was  found  on  photographic  plates 
taken  for  other  purposes,  and  is  one  of  the  few  examples  in 
which  comets  have  been  discovered  by  photography.  This 
comet  was  never  bright,  but  was  one  of  the  most  remarkable 
comets  ever  observed  in  the  extent  and  variety  of  its  activi- 
ties. It  was  well  situated  for  observation,  and  Barnard 
obtained  239  photographs  of  it  on  47  different  nights.  The 
material  which  went  into  the  tail  of  the  comet  was  often 
evolved  with  the  most  startUng  rapidity.     For  example,  on 


332       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  200 

the  30th  of  September,  in  the  early  part  of  the  night,  the 
comet  presented  an  almost  normal  appearance.  Before  the 
night  was  over,  the  tail  had  become  cyclonic  in  form  and 
was  attached  to  the  head,  which  then  was  small  and  star- 
like,  by  a  very  slender,  curved,  tapering  neck.  On  the 
"succeeding  night  the  material  that  then  constituted  the  tail 
was  entirely  detached  from  the  head.  On  October  15,  there 
was  another  large  outbreak  of  material  which  was  shown 
by  successive  photographs  to  be  swiftly  receding  from  the 
comet  (Fig.  126). 

Not  only  was  Morehouse's  comet  noteworthy  for  the  ex- 
traordinary activities  exhibited  by  its  tail,  but  it  changed  in 
brightness  in  a  very  remarkable  maimer.  It  was  generally 
considerably  below  the  Umits  of  visibility  with  the  vmaided 
eye,  but  now  and  then  it  would  flash  up,  without  apparent 
reason,  for  a  day  or  so  until  it  could  be  seen  very  faintly 
without  a  telescope.  While  a  number  of  larger  comets  have 
been  observed  in  recent  years,  no  other  has  given  evidence 
of  such  remarkable  changes  in  the  forces  that  produce  comets' 
tails,  and  no  other  has  exhibited  such  mysterious  variations 
in  brightness. 

201.  Halley's  Comet.  —  Halley's  comet  is  the  most  cele- 
brated one  in  all  the  history  of  these  objects.  It  is  named 
after  Halley,  not  because  he  discovered  it,  but  because  he 
computed  its  orbit  from  observations  made  in  1682  by  the 
methods  which  had  been  developed  by  his  friend  Newton. 
Halley  found  that  the  orbit  of  this  comet  was  almost  iden- 
tical with  the  orbits  of  the  comets  of  1607  and  1531.  He 
came  to  the  conclusion  that  these  various  comets  were  only 
different  appearances  of  the  same  one  which  was  ^evolving 
around  the  sun  in  a  period  of  about  75  years.  ^  ne  records 
of  comets  in  1456,  13D1,  1145,  and  1066  confirmed  this  view 
because  these  dates  differ  from  1682  by  nearly  integral  mul- 
tiples of  75  or  76  years.  From  his  computations  Halley  pre- 
dicted that  the  comet  would  appear  again  and  pass  its  peri- 
helion point  on  March  13,  1759. 


CH.  X,  201] 


COMETS  AND  METEORS 


333 


Fig.   126.  —  Morehouse's  comet,  Oct.  15,  1908.     Photographed  by  Barnard 
at  the  Yerkes  Observatory. 


334      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  201 

Many  of  Halley's  contemporaries  were  very  skeptical 
regarding  this  prediction.  The  law  of  gravitation  had  only 
recently  been  discovered  and  the  certainty  with  which  it 
had  been  estabhshed  was  not  yet  fully  comprehended. 
Halley  was  accused  by  skeptics  of  seeldng  notoriety  by 
making  a  prophecy  and  cleverly  putting  forward  the  date 
of  its  fulfillment  so  far  that  he  would  be  dead  before  his 
failure  became  known.  However,  before  the  75  years  had 
passed  away,  the  law  of  gravitation  had  become  so  firmly 
established,  and  the  mathematical  processes  employed  in 
astronomical  work  had  become  so  well  understood,  that 
astronomers,  at  least,  had  imphcit  faith  in  the  correctness 


Fig.   127.  —  The  orbit  of  Halley's  comet. 

of  Halley's  prediction,  although  since  its  last  appearance  the 
comet  had  been  invisible  for  the  lifetime  of  a  man  and  had 
gone  out  3,000,000,000  miles  from  the  sun  to  beyond  the 
orbit  of  Neptune.  There  was  great  popular  interest  in  the 
comet  as  the  date  for  its  return  approached.  It  actually 
passed  its  perihelion  within  one  month  of  the  time  predicted 
by  Halley.  The  slight  error  in  the  prediction  was  due  to 
the  imperfect  observations  of  its  positions  in  1682,  and  to 
the  perturbations  by  planets  which  were  then  unknown. 
This  was  the  first  verification  of  such  a  prediction ;  and  the 
definiteness  and  completeness  with  which  it  was  fulfilled 
had  been  entirely  unapproached  in  the  case  of  all  the 
prophecies  which  the  world  had  known  up  to  that  time. 
Halley's  comet  passed  the  sun  again  in  1835.     At  this 


CH.  X,  201]  COMETS  AND   METEORS  336 

time  -it  was  so  accurately  observed  that  its  subsequent  orbit 
could  be  computed  with  a  high  degree  of  precision.  If  it 
had  made  its  next  revolution  in  the  same  period  as  the  one 


Fig.  128. — Halley's  comet,  May  29,  1910.       Photographed  by  Barnard  at 
fhe  Yerkes  Observatory. 

ending  in  1835,  it  would  have  passed  its  perihelion  in  July, 
1912.  Instead  of  this,  it  passed  its  perihelion  on  April  19, 
1910.     The  perturbations  of  the  remote  planets  reduced  its 


336       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  201 


period  by  more  than  two  years.  The  most  accurate  com- 
putations of  its  orbit  and  predictions  of  the  time  of  its  re- 
turn were  made  by  Cowell  and  CromelUn,  of  Greenwich, 
who  missed  the  time  of  perihelion  passage  by  only  2.7  days. 
Their  computations  were  so  accurate  that  even  this  small 
discrepancy  could  not  be  the  result  of  accumulated  errors, 
and  they  beheve  that  the  comet  has  been  subject  to  some 

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Fig.   129.  —  The  relations  of  the  sun,  earth,  and  Halley's  comet  in  1910. 

unknown  forces.  Its  next  return  will  be  about  1985,  and 
Fig.  127  shows  the  position  in  its  orbit  for  various  epochs 
during  this  interval.  In  order  to  get  the  precise  time  of  its 
return,  it  will  be  necessary  to  take  into  account  the  pertur- 
bations of  the  planets. 

While  Halley's  comet  is  a  very  large  one  (Fig.  128),  its 
latest  appearance  was  somewhat  disappointing,  especially  to 
the  general  pubhc,  who  had  been  led  to  expect  that  it  would 


CH.  X,  202]  COMETS  AND   METEORS  337 

rival  the  sun  in  brightness.  One  of  the  reasons  for  the  dis- 
appointment was  that  the  earth  was  not  very  near  the  comet 
when  it  was  at  its  perihehon  where  it  was  brightest  and  had 
the  longest  tail.  The  relations  of  the  earth,  comet,  and 
sun  in  this  part  of  its  orbit  are  shown  in  Fig.  129,  drawn  by 
Barnard.  On  May  5,  the  length  of  the  comet's  tail  was 
37,000,000  miles.  On  May  18  the  comet  passed  between 
the  earth  and  the  sun  and  was  entirely  invisible  when  pro- 
jected on  the  sun's  disk.  This  shows  that  even  its  nucleus 
was  extremely  tenuous  and  transparent.  At  this  time  the 
earth  passed  through  at  least  the  outlying  part  of  its  tail. 
Neither  at  this  time  nor  at  any  other  did  the  comet  have 
any  sensible  influence  upon  the  earth.  On  the  whole,  it  was 
altogether  devoid  of  interesting  features. 

II.   Meteors 

202.  Meteors,  or  Shooting  Stars.  —  An  attentive  watch 
of  the  sky  on  almost  any  clear,  moonless  night  will  show  one 
or  more  so-called  "  shooting  stars."  They  are  little  flashes 
of  Ught  which  have  the  appearance  of  a  star  darting  across 
the  sky  and  disappearing.  Instead  of  being  actual  stars, 
which  are  great  bodies  like  our  sun,  they  are,  as  a  matter 
of  fact,  tiny  masses  so  small  that  a  person  could  hold  one 
in  his  hand.  Under  certain  circumstances  of  motion  and 
position,  they  dash  into  the  earth's  atmosphere  at  a  speed 
of  from  10  to  40  miles  per  second,  and  the  heat  generated 
by  the  friction  with  the  upper  air  vaporizes  or  burns  them. 
The  products  of  the  combustion  and  pulverization  slowly 
fall  to  the  earth  if  they  are  solid,  or  are  added  to  the  atmos- 
phere if  they  are  gaseous.  Since  it  is  misleading  to  call  them 
"  shooting  stars,"  they  will  always  be  called  "  meteors " 
hereafter. 

The  distances  of  meteors  were  first  determined  fn  1798 
by  Brandes  and  Benzenberg,  at  Gottingen.  They  made 
simultaneous  observations  of  them  from  positions  separated 
by  a  few  miles,  and  from  the  differences  in  their  apparent 


338       AN   INTRODUCTION   TO   ASTRONOMY    [ch.  x,  202 

directions  they  computed  their  altitudes  above  the  sur- 
face of  the  earth  (Art.  29).  Their  observations  and  those 
of  many  succeeding  astronomers,  among  whom  may  be 
mentioned  Denning,  of  England,  and  Olivier,  of  Virginia, 
have  shown  that  meteors  rarely,  if  ever,  become  visible  at 
,  altitudes  as  great  as  100  miles,  and  nearly  all  of  them  dis- 
appear before  they  have  descended  to  within  30  miles  of  the 
earth's  surface. 

The  velocity  with  which  a  meteor  enters  the  atmosphere 
can  be  found  by  determining  the  point  at  which  it  becomes 
visible,  the  point  at  which  it  disappears,  and  the  interval  of 
time  during  which  it  is  visible.  "  The  total  amount  of  light 
energy  given  out  by  a  meteor  can  be  determined  from  its 
apparent  brightness,  its  distance  from  the  observer,  and  the 
time  during  which  it  is  radiant.  The  energy  radiated  by  a 
meteor  has  its  source  in  the  heat  generated  by  the  friction 
of  the  meteor  with  the  earth's  atmosphere,  and  it  cannot 
exceed  the  kinetic  energy  of  the  meteor  when  it  entered 
the  atmosphere.  Suppose  all  the  kinetic  energy  of  a  meteor 
is  transformed  into  light.  This  assumption  is  not  strictly 
true,  but  it  will  be  approximately  true  for  matter  moving 
with  the  high  speed  of  a  meteor.  Then,  since  the  energy 
of  motion  of  a  body  is  one  half  its  mass  multiphed  by  the 
square  of  its  velocity,  the  mass  of  the  meteor  can  be  com- 
puted because  its  hght  energy  and  velocity  can  be  deter- 
mined directly  from  observations  by  the  methods  which 
have  just  been  described.  By  such  means  it  has  been  found 
that  ordinarily  the  masses  of  meteors  do  not  exceed  a  few 
tenths  of  an  ounce.  However,  the  observational  data  are 
difficult  to  determine  and  the  subject  has  received  relatively 
less  attention  than  it  deserves.  Consequently,  no  great 
rehance  should  be  placed  on  the  precise  numerical  results. 

203.  The  Number  of  Meteors.  —  If  a  person  scans  the 
sky  an  hour  or  so  and  finds  that  he  can  see  only  a  few  meteors, 
he  is  tempted  to  draw  the  conclusion  that  the  number  of 
them  which  strike  the  earth's  atmosphere  daily  is  not  very 


CH.  X,  204]  COMETS  AND   METEORS  339 

large.  He  bases  his  conclusion  mostly  on  the  fact  that  half 
of  the  celestial  sphere  is  within  his  range  of  vision,  but  a 
diagram  representing  the  earth  and  its  atmosphere  to  scale 
will  show  him  that  he  can  see  by  no  means  half  the  meteors 
which  strike  the  earth's  atmosphere.  As  a  matter  of  fact, 
he  can  see  the  atmosphere  over  only  a  few  square  miles  of 
the  earth's  surface. 

From  very  many  counts  of  the  number  of  meteors  which 
can  be  seen  from  a  single  place  during  a  given  time,  it  ha? 
been  computed  that  between  10  and  20  milUons  of  them 
strike  into  the  earth's  atmosphere  daily.  There  are  prob- 
ably several  times  this  number  which  are  so  small  that  they 
escape  observation.  Often  when  astronomers  are  working 
with  telescopes  they  see  faint  meteors  dart  across  the  field 
of  vision  which  would  be  quite  invisible  with  the  unaided  eye. 

Meteors  enter  the  earth's  atmosphere  from  every  direc- 
tion. The  places  where  they  strike  the  earth  and  the  veloci- 
ties of  their  encounter  depend  both  upon  their  own  veloci- 
ties and  also  upon  that  of  the  earth  around  the  sun.  The 
side  of  the  earth  which  is  ahead  in  its  motion  encounters 
more  meteors  than  the  opposite,  for  it  receives  not  only  those 
which  it  meets,  but  also  those  which  it  overtakes,  while  the 
part  of  the  earth  which  is  behind  receives  only  those  which 
overtake  it.  The  meridian  is  on  the  forward  side  of  the 
earth  in  the  morning  and  on  the  rearward  side  in  the  even- 
ing. It  is  found  by  observation  that  more  meteors  are  seen 
in  the  morning  than  in  the  evening,  and  that  the  relative 
velocities  of  impact  are  greater. 

204.  Meteoric  Showers.  - —  Occasionally  unusual  num- 
bers of  meteors  are  seen,  and  then  it  is  said  that  there  is  a 
meteoric  shower.  There  have  been  a  few  instances  in  which 
meteors  were  so  numerous  that  they  could  not  be  counted, 
but  usually  not  more  than  one  or  two  appear  in  a  minute. 

At  the  time  of  a  meteoric  shower  the  meteors  are  not 
only  more  numerous  than  usual,  but  a  majority  of  them 
move  so  that  when  their  apparent  paths  are  projected  back- 


340        AN    INTRODUCTION    TO   ASTKUJn  Uivi  i     lua.  a.  ^u^r 

ward,  they  pass  through,  or  very  near,  a  point  in  the  sky. 
This  point  is  called  the  radiant  point  of  the  shower,  for  the 
meteors  all  appear  to  radiate  from  it.     A  number  of  meteor^ 
trails  which  clearly  define  a  radiant  point   are  shown  in 
Fig.  130. 

The  most  conspicuous  meteoric  showers  occur  on  Novem- 
ber 15  and  November  24  yearly.  The  former  have  their 
radiant  in  Leo,  within  the  sickle,  and  are  called  the  Leonids. 


Fig.  130.  —  Meteor  trails  defining  a  radiant  point  (Olivier). 

From  the  position  of  this  constellation  (Arts.  82,  93),  it 
follows  that  they  can  be  seen  only  in  the  early  morning  hours. 
The  latter  have  their  radiant  in  Andromeda,  and  are  called 
the  Andromids.  They  can  be  seen  only  in  the  early  part 
of  the  night.  The  Leonids  and  Andromids  are  not  equally 
numerous  every  year.  Great  showers  of  the  Leonids  oc- 
curred in  1833  and  1866,  and  less  remarkable  ones,  though 
greater  than  the  ordinary,  from  1898  to  1901.  The  Andro- 
mids appear  in  unusual  numbers  every  thirteen  years. 
Besides  these  meteoric  showers,  according  to  Denning, 


CH.  X,  206] 


COMETS  AND   METEORS 


341 


nearly  3000  other  less  conspicuous  ones  have  been  found. 
The  Perseids  appear  for  a  week  or  more  near  the  middle  of 
August,  the  Lyrids  on  or  about  April  20,  the  Orionids  on  or 
about  October  20,  etc. 

205.  Explanation  of  the  Radiant  Point.  —  In  1834  Olm- 
sted showed  that  the  apparent  radiation  of  meteors  from  a 
point  is  due  to  the  fact  that  they  move  in  parallel  lines, 
and  that  we  see  only  the  projection  of  their  motion  on  the 
celestial  sphere.  Thus,  in  Fig.  131,  the  actual  paths  of  the 
meteors  are  AB,  but  their  apparent  paths  as  seen  by  an 
observer  at  0  are  AC.    When  these  lines  are  all  continued 


*^1  \ 


Fig.   131.  —  Explanation  of  the  radiant  point  of  meteors. 


.-e. 


backward,  they  meet  in  the  point  which  is  in  the  direction 
from  which  the  meteors  come. 

It  follows  that  the  meteors  which  give  rise  to  the  meteoric 
showers  are  moving  in  vast  swarms  along  orbits  which  inter- 
sect the  orbit  of  the  earth.  When  the  earth  passes  through 
the  point  of  intersection,  it  encounters  the  meteors  and  a 
shower  occurs.  Thus,  the  orbit  of  the  Leonids  touches  the 
orbit  of  the  earth  at  the  point  which  the  earth  occupies  bn 
November  14.  In  this  case  the  earth  meets  the  meteors 
(Fig.  132),  while  the  Andromids  overtake  the  earth. 

206.  Connection  between  Comets  and  Meteors.  —  The 
fact  that  the  volatile  material  of  which  comets'  tails  are 
composed  gradually  becomes  exhausted,  after  which  the 
comets  themselves  become  invisible,  and  the  fact  that 
meteoric  showers  are  due  to  wandering  swarms  of  small 


342       AN   INTRODUCTION    TO  ASTRONOMY    [ch.  x,  zue 


particles  which  revolve  around  the  sun  in  elongated  ellip- 
tical orbits,  suggest  the  hypothesis  that  comets  and  meteors 
are  related.  The  hypothesis  is  confirmed  and  virtually 
proved  by  the  identity  of  the  orbits  of  certain  meteoric 
swarms  and  comets. 

In  1866  Schiaparelli  showed  that  the  August  meteors 
move  in  the  same  orbit  as  Tuttle's  comet  of  1862.  That  is, 
in  addition  to  the  comet,  which  is  a  member  of  Saturn's 
family,  there  are  many  other  small  bodies  (meteors)  travel- 
ing in  the  same  orbit.  In  1867  Leverrier  found  that  the 
Leonids  move  in  the  same  orbit  as  Tempel's  comet  of  1866, 
while  Weiss  showed  that  the  meteors  of  April  20  and  the 


Fig.   132.  —  Orbit  of  the  Leonid  meteors. 

comet  of  1861  move  in  the  same  orbit,  and  that  the  paths 
of  the  Andromids  and  Biela's  comet  were  likewise  the  same. 
It  has  recently  been  claimed  that  the  Aquarid  meteors  of 
early  May  have  an  orbit  almost  identical  with  that  of 
Halley's  comet. 

While  it  is  not  possible  to  be  certain  as  to  the  origin  of 
comets,  the  history  of  their  later  evolution  and  final  end  is 
tolerably  clear.  The  elongated  orbits  in  which  they  may 
have  originally  moved  are  reduced  when  they  are  captured 
by  the  planets.  Their  periods  of  revolution  are  subsequently 
shorter,  their  volatile  material  wastes  away  in  the  form  of 
tails,  and  the  remaining  material  is  scattered  along  their 
orbits  by  the  dispersive  forces  to  which  they  are  subject. 


CH.  X,  208]  COMETS  AND   METEORS  343 

If  these  orbits  cross  the  orbit  of  a  planet,  the  remains  of  the 
comets  are  gradually  sWept  up  by  the  larger  body.  If  an  orbit 
of  a  comet  does  not  originally  cross  the  orbit  of  a  planet, 
the  perturbations  of  the  planets  will,  in  general,  in  the  course 
of  time,  cause  it  to  do  so.  The  result  will  be  that  the  planets 
sweep  up  more  and  more  of  the  remains  of  disintegrated 
comets  and  undergo  a  gradual  growth  in  this  manner. 

207.  Effects  of  Meteors  on  the  Solar  System. —  The 
most  obvious  effect  of  the  numerous  meteors  which  swarm 
in  the  solar  system  is  a  resistance  both  to  the  rotations  and 
the  revolutions  of  all  the  bodies.  As  was  stated  in  Art.  45, 
the  effects  of  meteors  upon  the  rotation  of  the  earth  are  at 
present  exceedingly  slight,  and  it  is  very  probable  that  their 
influences  upon  the  rotations  of  the  other  members  of  the 
system  are  also  inappreciable.  A  retardation  in  the  trans- 
latory  motion  of  a  body  causes  its  orbit  to  decrease  in  size. 
Hence,  so  far  as  the  meteors  affect  the  planets  in  this  way, 
they  cause  them  continually  to  approach  the  sun. 

Another  effect  of  meteors  upon  the  members  of  the  solar 
system  is  to  increase  their  masses  by  the  accretion  of  matter 
which  may  have  come  originally  from  far  beyond  the  orbit 
of  Neptune.  As  the  masses  of  the  sun  and  planets  in- 
crease, their  mutual  attractions  increase  and  the  orbits  of 
the  planets  become  smaller.  Looking  backward  in  time,  we 
are  struck  by  the  possibility  that  the  accretion  of  meteoric 
matter  may  have  been  more  rapid  in  former  times,  and  that 
it  may  have  been  an  important  factor  in  the  growth  of  the 
planets  from  much  smaller  bodies. 

208.  Meteorites.  —  Sometimes  bodies  weighing  from  a 
few  pounds  up  to  several  hundred  pounds,  or  even  a  few 
tons,  dash  into  the  earth's  atmosphere,  glow  brilliantly  from 
the  heat  generated  by  the  friction,  roar  like  a  waterfall, 
occasionally  produce  violent  detonations,  and  end  by  falling 
on  the  earth.  Such  bodies  are  called  meteorites,  siderites,  or 
aerolites. 

About  two  or  three  meteorites  are  seen  to  fall  yearly ;  but. 


344       AN   INTRODUCTION  TO  ASTRONOMY    [ch.  x,  vsus 

since  a  large  part  of  the  earth  is  covered  with  water  or  is 
uninhabited  ior  other  reasons,  it  is  probable  that  in  all 
at  least  100  strike  the  earth  annually.  The  outside  of  a 
meteorite  during  its  passage  through  the  air  is  subject  to 
intense  and  sudden  heating,  and  the  rapid  expansion  of  its 
surface  layers  often  breaks  it  into  many  fragments.  The 
surface  is  fused  and  on  striking  cools  rapidly.  The  result  is 
that  it  has  a  black,  glossy  structure,  usually  with  many 
small  pits  where  the  less  refractive  material  has  been  melted 


Fig.   133.  —  Stony  meteorite  which  fell  at  Long  Island,  Kansas ;  weight, 
700  pounds  (Farrington). 

out.  Since  meteors  pass  entirely  through  the  atmosphere  in 
a  few  seconds,  only  their  surfaces  give  evidence  of  the  ex- 
tremes of  heat  and  pressure  to  which  they  have  been  sub- 
jected in  their  final  flight. 

Most  meteors  are  composed  of  stone,  though  it  is  often 
mixed  with  some  metallic  iron.  Even  where  pure  iron  is  not 
present,  some  of  its  compounds  are  usually  found.  About 
three  or  four  out  of  every  hundred  are  nearly  pure  iron 
with  a  little  nickel.  All  together  about  30  elements  which 
occur  elsewhere  on  the  earth  have  been  found  in  meteorites, 
but  no  strange  ones.  Yet  in  some  respects  their  structure 
is  quite  different  from  that  of  terrestrial  substances.    They 


CH.  X,  209] 


COMETS  AND  METEORS 


345 


have  peculiar  crystals,  they  show  but  little  oxidation  and 
no  action  of  water,  and  they  contain  in  their  interstices  rela- 
tively large  quantities  of  occluded  gases,  some  of  which  are 


Fig.  134.  —  Iron  meteorite  from  Canon  Diablo,  Arizona ; 
pounds  (Farrington) . 


weight,  265 


combustible.  According  to  Farrington,  some  meteors  give 
evidence  of  fragmentation  and  recementation,  others  show 
faulting  (fracture  and  shding  of  one  surface  on  another)  with 


Fig.  135.  —  Durango,  Mexico.     Meteorite  showing  peculiar  crystallization 
characteristic  of  certain  meteorites  (Farrington). 

recementation,  and  others,  veins  where  foreign  material  has 
been  slowly  deposited. 

209.  Theories  respecting  the  Origin  of  Meteorites.  —  If 
it  were  known  that  meteorites  are  but  meteors  which  are  so 
large  that  they  reach  the  earth  before  they  are  completely 


346       AN    INTRODUCTION   TO   ASTRONOMY    [ch.  x,  209 

oxidized  and  pulverized,  we  might  justly  conclude  that  they 
are  probably  the  remains  of  disintegrated  comets.  This 
would  enable  us  to  learn  certain  things  about  comets  which 
cannot  be  settled  yet.  But  no  meteorite  is  known  certainly 
to  have  been  a  member  of  any  meteoric  swarm.  However, 
two  meteorites  have  fallen  during  the  time  of  meteoric 
showers,  one  in  France,  at  the  time  of  the  Lyrids  in  1905, 
and  the  other  in  Mexico,  just  before  the  Andromids  in  1885. 

The  structure  of  some  meteorites  is  more  like  that  of  lava 
from  deep  volcanoes  than  anything  else  found  on  the  earth. 
An  old  theory  was  that  they  have  been  ejected  by  volcanic 
explosions  from  the  moon,  planets,  or  perhaps  the  sun. 
This  theory  would  account  for  some  of  their  characteristics, 
and  would  explain  why  they  contain  only  familiar  elements, 
at  least  if  the  other  bodies  of  the  solar  system  contain  only 
those  found  on  the  earth ;  but  it  does  not  at  all  explain  the 
fragmentation,  faulting,  and  veins,  for  forces  great  enough 
to  produce  ejections  would  scarcely  be  found  without  heat 
enough  to  produce  at  least  fusion. 

Chamberlin  has  maintained  that  meteorites  may  be  the 
debris  of  bodies,  perhaps  of  planetary  dimensions,  which 
have  been  broken  up  by  tidal  strains  when  they  have  passed 
some  larger  rhass  within  Roche's  limit.  When  suns  pass  by 
other  suns,  it  is  probable  that  at  rare  intervals  they  pass  so 
near  each  other  that  their  planets  (if  they  have  any)  are 
broken  up.  More  rarely,  the  suns  themselves  may  be  dis- 
integrated. Indeed,  this  may  be  the  origin  of  all  cometary 
and  meteoric  matter.  Whether  it  is  or  not,  there  is  here 
a  possibility  of  disintegration  which  must  be  taken  into 
account  in  any  theory  of  cosmical  evolution. 

The  present  desiderata  are  more  accurate  determinations 
of  comets'  orbits  to  find  whether  any  of  them  are  really 
hyperbolic,  more  accurate  determinations  of  the  velocities  of 
meteors  to  find  whether  they  ever  come  into  our  system  on 
parabohc  or  hyperbolic  orbits,  and  finally  the  answer  to  the 
question  whether  meteors  and  meteorites  are  really  related. 


CH.  X,  2U9J  CUMET8   AMD    METEORS  347 

The  suggestion  that  a  meteorite  may  be  a  fragment  of' a 
world  which  was  disrupted  before  the  origin  of  the  earth 
makes  some  demands  on  the  imagination,  but  it  seems  no 
more  incredible  to  us  than  seemed  the  suggestion  to  our 
predecessors  a  century  ago  that  great  mountains  have  been 
utterly  destroyed  by  the  rains  and  snows  and  winds. 

XIV.  QUESTIONS 

1.  What  observations  would  prove  that  comets  are  not  in  the 
earth's  atmosphere,  as  the  ancients  supposed  they  were  ? 

2.  Suppose  two  small  masses  are  moving  around  the  sun  in  the 
same  elongated  orbit,  but  that  one  is  somewhat  ahead  of  the  other. 
How  will  their  distance  apart  vary  with  their  position  in  their  orbit 
(use  the  law  of  areas)  ?  Does  this  suggest  an  explanation  of  the 
variations  in  the  dimensions  of  comets'  heads  ? 

3.  The  velocity  of  a  comet  moving  in  a  paraboUc  orbit  is  in- 
versely as  the  square  root  of  its  distance  from  the  sun.  At  the  dis- 
tance of  the  earth  a  comet  has  a  velocity  of  about  25  miles  per 
second.  What  is  the  distance  between  the  comets  of  1843  and  1882 
when  they  are  100,000  astronomical  units  from  the  sun  ? 

4.  Suppose  the  particles  of  which  a  comet  is  composed  have 
almost  exactly  the  same  perihelion  point  but  somewhat  different 
aphehon  points.  How  would  the  dimensions  of  the  comet  vary 
witfcits  position  in  its  orbit  ? 

5.  By  means  of  Kepler's  third  law  compute  the  period  of  a 
comet  whose  aphehon  point  is  at  a  distance  of  140,000  astronomical 
units,  which  is  about  half  the  distance  of  the  nearest  known  star. 

6.  What  objections  are  there  to  the  theory  that  originally  all 
comets  had  an  aphehon  distance  equal  to  that  of  Neptune,  and  that 
the  orbits  of  some  have  been  increased  and  others  diminished  by  the 
action  of  the  planets  ? 

7.  On  the  repulsion  theory  should  a  comet's  tail  be  equally  long 
when  it  is  approaching  the  sun  and  when  it  is  receding  ? 

8.  Draw  the  diagram  mentioned  in  the  first  paragraph  of  Art.  203. 

9.  Count  the  number  of  meteors  you  can  observe  in  an  hour  on 
some  clear,  moonless  night; 

'  10.   If  possible,  observe  the  Leonid  or  Andromid  meteors. 
1 1 .   Make  a  list  of  the  fairly  well-explained  cometary  phenomena, 
and  of  those  for  which  no  satisfactory  theory  exists. 


Fig.   136.  —  The  tower  telescope  of  the  solar  observatory  of  the  Carnegie 

Institution  of  Washington,  Pasadena,  California. 

348 


CHAPTER  XI 
THE    SUN 

I.   The  Sun's  Heat 

210.  The  Problem  of  the  Sun's  Heat.  —  The  light  and 
heat  radiated  by  the  sun  are  essential  for  the  existence  of  life 
on  the  earth,  and  consequently  the  question  of  the  source 
of  the  sun's  energy,  how  long  it  has  been  suppUed,  and  how 
long  it  will  last  are  of  vital  interest.  Not  only  are  these 
questions  of  importance  because  the  sun  is  the  dominant 
member  of  the  solar  system,  governing  the  motions  of  the 
planets  and  illuminating  and  heating  them  with  its  abun- 
dant rays,  but  also  because  the  sun  is  a  star,  and  the  only 
one  of  the  hundreds  of  milhons  in  the  sky  which  is  so  near 
that  its  surface  can  be  studied  in  detail. 

Obviously  the  first  thing  to  do  in  stud3dng  the  heat  of  the 
sun  is  to  measure  the  amount  received  from  it  by  the  earth ; 
then,  the  amount  which  the  sun  radiates  can  be  computed. 
The  amount  of  heat  given  out  by  the  sun  gives  the  basis  for 
determining  its  temperature.  Then  naturally  follows  the 
question  of  the  origin  of  the  sun's  heat.  The  answers  to 
these  questions  are  of  great  importance  in  considering  the 
the  evolution  of  the  solar  system  and  the  stars. 

211.  The  Amount  of  radiant  Energy  received  by  the 
Earth  from  the  Stm.  —  Light  is  a  wave  motion  in  the  ether 
whose  wave  lengths  vary  from  about  sshm  of  an  inch,  in 
the  violet,  to  about  10:500  of  an  inch,  in  the  red.  Radiant 
heat  differs  from  Hght  physically  only  in  that  its  waves  are 
longer.  The  circumstance  that  human  eyes  are  sensitive 
to  ether  waves  of  certain  lengths  and  not  to  those  that  are 
longer  or  shorter  is,  of  course,  of  no  importance  in  discussing 

349 


350      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  211 

the  physical  question  of  the  sun's  heat.  Consequently,  in 
the  problem  of  solar  radiation  rays  of  all  wave  lengths  are 
included,,  and  together  they  constitute  the  radiant  energy 
emitted  by  the  sun. 

Physicists  have  devised  various  methods  of  measuring 
the  amount  of  energy  received  from  a  radiating  source. 
In  applying  them  to  the  problem  of  determining  the  amount 
of  energy  received  from  the  sun  the  chief  difficulty  consists 
in  making  correct  allowance  for  the  absorption  of  hght  and 
heat  by  the  earth's  atmosphere.  The  best  results  have  been 
obtained  by  making  simultaneous  measurements  from  near 
sea  level,  from  the  summits  of  lofty  mountains,  and  from 
balloons.  Langley  measured  the  intensity  of  solar  radi- 
ation at  the  top  of  Mount  Whitney,  14,887  feet  above  the 
sea,  and  at  its  base.  He  arrived  at  the  conclusion  that  40 
per  cent  of  the  rays  striking  the  atmosphere  perpendicularly, 
when  it  is  free  from  clouds,  are  absorbed  before  they  reach 
the  surface  of  the  earth ;  later  investigations  have  reduced 
this  estimate  to  35  per  cent.  The  work  initiated  by  Langley 
has  been  continued  most  successfully  by  Abbott,  Fowle,  and 
Aldrich,  and  they  find  that  the  rate  at  which  radiant  energy 
of  all  wave  lengths  is  received  by  the  earth  from  the  sun  at 
the  outer  surface  of  our  atmosphere  when  the  sun  is  at  its 
mean  distance  is,  in  terms  of  mechanical  work,  1.51  horse 
power  per  square  yard. 

The  earth  intercepts  a  cyUnder  of  rays  from  the  sun  whose 
cross  section  is  equal  to  a  circle  whose  diameter  equals  the 
diameter  of  the  earth.  The  area  of  this  circle  is,  therefore, 
7rr2,  where  r  equals  3955  X  1760  =  6,960,000  yards.'  Hence 
the  rate  at  which  solar  energy  is  intercepted  by  the  whole 
earth  is  in  round  numbers  230,000,000,000,000  horse  power. 

In  the  evolution  of  life  upon  the  earth  the  sun  has  been  as 
important  a  factor  as  the  earth  itself.  Consequently,  geolo- 
gists and  biologists  have  a  deep  interest  in  the  sun,  and  par- 

'  The  mean  radius  of  the  earth  is  3955  miles  and  there  are  1760  yards  in 
a  mile. 


CH.  XI,  212]  THE    SUN  351 

ticularly  in  the  question  whether  or  not  its  rate  of  radiation 
is  constant.  It  has  long  been  supposed  that  probably  the 
sun  is  slowly  cooUng  ofE  and  that  the  light  and  heat  received 
from  it  are  gradually  diminishing,  but  it  was  a  ■distinct  sur- 
prise when  Langley  and  Abbott  found  that  its  rate  of  radi- 
ation sometimes  varies  in  a  few  days  by  as  much  as  10 
per  cent.  If  a  change  of  this  amount  in  the  rate  of  radiation 
of  the  sun  were  to  persist  indefinitely,  the  mean  temperature 
of  the  earth  would  be  changed  about  13°  Fahrenheit ;  but 
a  variation  of  10  per  cent  for  only  a  few  days  has  no  im- 
portant effect  on  the  chmate.  Abbott,  Fowle,  and  Aldrich 
have  continued  the  investigation  of  this  question,  and  by 
making  observations  simultaneously  in  Algiers,  in  Washing- 
ton, and  in  CaUfornia,  so  as  to  ehminate  the  effects  of  local 
and  transitory  atmospheric  conditions,  they  have  firmly 
estabhshed  the  reality  of  small  and  rapid  variations  in  the 
sun's  rate  of  radiation. 

The  question  of  variation  in  the  amount  of  energy  received 
from  the  sun  can  also  be  considered  in  the  light  of  geological 
evidence.  The  fossils  preserved  in  the  rocks  of  all  geological 
ages  prove  that  there  has  been  an  unbroken  life  chain  upon 
the  earth  for  many  tens .  of  millions  of-  years.  This  means 
that  during  all  this  vast  period  of  time  the  temperature  of 
the  earth  has  been  neither  so  high  nor  so  low  as  to  destroy 
all  Uf e.  Moreover,  the  record  is  clear  that,  in  spite  of  glacial 
epochs  and  intervening  warmer  eras,  the  temperature  changes 
have  not  been  very  great,  and  there  is  no  evidence  of  a  pro- 
gressive coohng  of  the  sun. 

212.  Sources  of  the  Energy  used  by  Man.  —  One  of  the 
eariiest  extensive  sources  of  energy  for  mechanical  work  used 
by  man  was  the  wind.  It  has  tifrned,  and  still  turns,  mil- 
lions of  windmills  for  driving  machinery  or  pumping  water. 
Until  the  last  few  decades  it  moved  nearly  all  of  the  ocean- 
borne  commerce  of  the  w;hole  world,  and  it  is  still  an  impor- 
tant factor  in  shipping.  But  that  part  of  the  energy  of  the 
wind  which  is  used  is  an  insignificant  fraction  of  all  that 


352      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  212 

exists.  For  example,  if,  in  a  breeze  blowing  at  the  rate  of 
20  miles  an  hour,  all  the  energy  in  the  air  crossing  an  area 
100  feet  square  perpendicular  to  its  direction  of  motion  were 
used,  it  would  do  about  560  horse  power  of  work. 

What  is  the  origin  of  the  energy  in  the  wind?  The  sun 
warms  the  atmosphere  over  the  equatorial  regions  of  the 
earth  more  than  that  over  the  higher  latitudes,  and  the 
resulting  convection  currents  constitute  the  wind.  Con- 
sequently, all  the  energy  in  every  wind  that  blows  came  orig- 
inally from  the  sun. 

Another  source  of  energy  which  has  been  of  great  practical 
value  is  water  power.  The  source  of  this  energy  is  also  the 
sun,  because  the  sun's  heat  evaporates  the  water  and  raises 
it  into  the  air  a  half  mile  or  more,  the  winds  carry  part  of 
it  out  over  the  land,  where  it  falls  as  rain  or  snow,  and  in 
descending  again  to  the  ocean  it  may  now  and  then  plunge 
over  a  precipice,  where  its  energy  can  be  utilized  by  men. 
Amazing  as  are  the  figures  for  such  great  waterfalls  as 
Niagara,  they  give  but  a  faint  idea  of  the  enormous  work  the 
sun  has  done  in  raising  water  into  the  sky,  and  the  equally 
great  amount  of  work  the  water  does  in  falhng  back  to  the 
earth.  During  a  heavy  rain  an  inch  of  water  may  fall. 
An  inch  of  water  on  a  square  mile  weighs  over  60,000  tons. 
In  the  eastern  half  of  the  United  States,  where  the  annual 
rainfall  is  about  35  inches,  every  year  over  2,000,000  tons  of 
water  fall  on  each  square  mile  from  a  height  of  half  a  mile  or 
more. 

The  great  modern  source  of  energy  for  mechanical  work  is 
coal.  The  coal  has  formed  from  vegetable  matter  which 
accumulated  in  peat  beds  ages  and  ages  ago.  Consequently, 
the  immediate  source  of  its  energy  is  the  plants  out  of  which 
it  has  developed.  But  the  plants  obtained  their  energy  from 
the  sun.  In  millions  of  tiny  cells  the  sun's  energy  broke 
up  the  carbon  dioxide  which  they  inhaled  from  the  atmos- 
phere; then  the  oxygen  was  exhaled  and  the  carbon  was 
stored  up  in  their  tissues.     When  a  plant  is  burned,  as  much 


CH.  XI,  213]  THE    SUN  353 

energy  is  developed  and  given  up  again  as  the  sun  put  into 
it  when  it  grew. 

Thus  it  is  seen  that  all  the  great  sources  of  energy  can  be 
traced  back  to  the  sun;  it  is  true  of  the  minor  ones  also. 
One  naturally  inquires  whether  these  sources  of  energy  are 
perpetual.  The  winds  will  certainly  continue  to  blow  and 
the  rains  to  descend  as  long  as  the  earth  and  sun  exist  in 
their  present  conditions,  but  the  coal  and  petroleum  will 
eventually  be  exhausted.  They  will  last  several  centuries 
and  perhaps  a  few  thousand  years.  This  period  seems  long 
compared  to  the  Hfetime  of  an  individual,  or  perhaps  of  a 
nation,  but  it  is  only  a  minute  fraction  of  the  time  during 
which  our  successors  will  probably  occupy  the  earth.  It 
follows  that  they  will  be  compelled  to  depend  upon  sources 
of  energy  at  present  but  little  utilized.  Perhaps  some  great 
benefactor  of  mankind  will  discover  a  means  of  putting  to 
direct  use  the  enormous  quantities  of  energy  which  the  sun 
is  now  sending  to  the  earth.  At  present  we  are  depending 
on  that  infinitesimal  residue  of  the  energy  which  the  earth 
received  in  earher  geological  times  and  which  has  been 
stored  up  and  preserved  in  petroleum  and  coal. 

213.  The  Amount  of  Energy  radiated  by  the  Sun. — 
The  earth  as  seen  from  the  sun  subtends  an  angle  of  only 
17".6.  That  is,  its  apparent  area  is  about  tV  the  greatest 
apparent  area  of  Venus  as  seen  from  the  earth.  A  glance 
at  Venus  will  show  that  this  is  an  exceedingly  small  part  of 
the  whole  celestial  sphere.  Since  the  little  earth  at  a  dis- 
tance of  93,000,000  of  miles  receives  the  enormous  quantity 
of  heat  given  in  Art.  211,  it  follows  that  the  amount  which 
is  radiated  by  the  sun  must  be  inconceivable.  It  can  be 
brought  within  the  range  of  our  understanding  only  by  con- 
templating some  of  the  things  it  might  do. 

The  energy  radiated  per  square  yard  from  the  sun's  surface 

is  equivalent  to  70,000  horse  power.     This  amount  of  heat 

energy  would  melt  a  layer  of  ice  2200  feet  thick  every  hour 

all  over  the  surface  of  the  sun ;  and  it  would  melt  a  globe  of 

2a 


354      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  213 

ice  as  large  as  the  earth  in  2  hours  and  40  minutes.  Less 
than  one  two-bilhonth  of  the  energy  poured  forth  by  the  sun 
is  intercepted  by  the  earth,  and  less  than  ten  times  this 
amount  by  all  the  planets  together ;  the  remainder  travels 
on  through  the  ether  to  the  regions  of  the  stars  at  the  rate 
of  186,000  miles  per  second. 

214.   The  Temperature  of  the  Sun.  —  Stefan's  law  (Art. 
172)  that  a  black  body  radiates  as  the  fourth  power  of  its 
absolute  temperature,  gives   a  basis   for   determining  the 
temperature  of  a  body  whose  rate  of  radiation  is  known. 
While  the  sun  is  probably  not  an  ideal  radiator,  such  as  is 
contemplated  in  the  statement  of  Stefan's  law,  and  while 
it  radiates  from  layers  at  various  depths  below  its  surface, 
with  the  upper  layers  absorbing  part  of  the  energy  coming 
from  the  lower,  yet  an  approximate  idea  of  the  temperature 
of  its  radiating  layers  can  be  obtained  from  its  rate  of  radi- 
ation.    On  using  Stefan's  law  as  a  basis  for  computation,  it 
is  foimd  that  the  temperature  of  the  radiating  layers  of  the 
~-sun  is  at  least  10,000°  Fahrenheit.     Or,  it  would  be  more 
accurate  to  say  that  an  ideal  radiating  surface  at  this  tem- 
perature would  have  the  same  rate  of  radiation  as  the  sun, 
and  since  the  sun  is  not  a  perfect  radiator,  its  temperature 
is  probably  still  higher.     This  temperature  is  several  thou- 
sand degrees  higher  than  has  been  obtained  in  the  most 
efficient  electrical  furnaces,  and  is  far  beyond  that  required 
to  melt  or  vaporize  any  known  terrestrial  substance;  yet, 
the  temperature  of  the  interior  of  the  sun  is  undoubtedly 
far  higher. 

Another  method  of  determining  the  temperature  of  the 
sun  is  from  the  proportion  of  energy  of  different  wave  lengths 
which  it  radiates.  A  body  of  low  temperature  radiates 
relatively  a  large  amount  of  red  Ught  and  a  small  amoimt  of 
blue  fight.  As  the  temperature  rises  the  relative  proportion 
of  blue  fight  increases.  The  uncertainties  in  the  results  ob- 
tained by  this  method  of  determining  the  temperature  of 
the  sun  arise,  in  the  first  place,  from  the  fact  that,  at  the 


CH.  XI,  215]  THE    SUN  355 

best,  it  is  not  very  precise,  and,  in  the  second  place,  from  the 
fact  that  both  the  sun's  and  the  earth's  atmospheres  absorb 
very  unequally  radiant  energy  of  various  wave  lengths. 
After  making  the  necessary  allowances  for  the  absorption, 
the  results  obtained  by  this  method  confirm  those  found  by 
the  other. 

There  have  been  a  nmnber  of  other  methods  of  obtaining 
the  temperature  of  the  sun  from  the  time  of  Newton,  but 
most  of  them  have  rested  on  physical  principles  which  are 
unsound,  and  in  some  cases  they  have  led  to  most  extravagant 
results. 

215.  The  Principle  of  the  Conservation  of  Energy.  —  Be- 
fore taking  up  the  question  of  the  origin  of  the  sun's  heat, 
it  is  advisable  to  consider  the  principle  of  the  conservation 
of  energy.  It  is  comparable  in  importance  and  generality 
to  the  principle  of  the  conservation  (indestructibiUty)  of 
matter.  It  was  once  supposed  that  when  inflammable  ma- 
terial, as  wood,  is  burned,  it  is  utterly  annihilated.  But  it 
has  been  known  for  about  150  years  that  if  the  ashes,  the 
smoke,  and  the  gases  produced  by  the  combustion  were  all 
gathered  up  and  weighed  in  a  vacuum,  their  weight  would 
exactly  equal  that  of  the  original  wood  together  with  the 
oxygen  which  united  with  it  in  burning. 

Similarly,  it  was  supposed  imtil  after  1840  that  energy 
might  be  destroyed  as  well  as  transformed.  For  example, 
it  was  supposed  that  the  energy  lost  by  friction  ceased  to 
exist.  But  it  had  been  noted  that  friction  produced  heat, 
and  heat  was  known  to  be  equivalent  to  mechanical  energy, 
for  it  had  been  turned  into  work,  for  example,  by  means 
of  the  steam  engine.  It  does  not  seem  now  to  have  been 
a  large  step  to  have  conjectured  that  the  heat  produced  by 
the  friction  is  exactly  equivalent  to  the  energy  lost.  But 
many  elaborate  experiments  were  required  (made  mostly  by 
Mayer  and  Joule)  to  prove  the  correctness  of  this  conjecture 
and  to  lead  to  the  generaUzation,  now  universally  accepted, 
that  the  total  amount  of  energy  in  the  universe  is  always  the 


356      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  :ii6 

same.  This  is  one  of  the  most  far-reaching  principles  of  sci- 
ence, and,  like  the  law  of  gravitation,  is  involved  in  every 
phenomenon  in  which  there  is  motion  of  matter. ' 

The  energy  of  a  body  as  used  in  the  principle  of  the  con- 
servation of  energy  means  both  its  energy  of  motion  (kinetic 
energy)  and  also  its  energy  of  position,  or  the  power  it  may 
have  of  doing  work  because  of  its  position  (potential  energy). 
It  is  the  smn  of  the  potential  and  kinetic  energies  of  the 
universe  which  is  constant.  Since  energy  may  be  in  a  radi- 
ant form  and  in  transit  from  one  body  to  another,  or  from 
a  body  out  into  endless  space,  the  principle  holds  only  when 
the  energy  which  is  in  the  ether  is  also  included. 

216.  The  Contraction  Theory  of  the  Sun's  Heat.  —  The 
mutual  attractions  of  the  particles  of  which  the  sun  is  com- 
posed tend  to  cause  it  to  contract.  A  contraction  of  the  sun 
would  be  equivalent  to  a  fall  of  all  of  its  particles  toward 
its  center.  If  they  should  fall  the  whole  distance  one  at  a 
time,  they  would  generate  a  certain  amount  of  heat  upon 
their  impacts.  If  they  should  fall  simultaneously,  first  a 
fraction  of  the  distance  and  then  another,  the  same  total 
amount  of  heat  would  be  generated.  It  might  be  supposed 
without  computation  that  an  enormous  contraction  would 
be  necessary  in  order  to  produce  enough  heat  to  change 
appreciably  the  temperature  of  the  sun. 

The  effect  of  the  sun's  contraction  can  be  considered  more 
exactly  in  terms  of  energy.  The  sun  in  an  expanded  con- 
dition would  have  more  potential  energy  with  respect  to 
the  force  of  gravitation  than  if  it  were  contracted,  because 
work  would  be  done  on  it  by  gravitation  in  changing  it  from 
the  first  state  to  the  second.  Therefore  the  kinetic  energy, 
or  temperature,  of  the  sun  must  rise  on  its  contraction.  It 
is  analogous  to  a  falling  body.  The  higher  it  is  above  the 
surface  of  the  earth,  the  greater  its  potential  energy ;  the 
farther  it  falls,  the  more  potential  energy  it  loses  and  the 
more  kinetic  energy  it  acquires. 

The  problem  is  to  determine  whether  the  contraction  of 


CH,  XI,  216]  THE    SUN  357 

the  sun  might  not  supply  it  with  heat  to  take  the  place  of 
that  which  it  radiates  so  lavishly.  With  the  insight  of 
genius,  Helmholtz  saw  the  nature  of  the  question  and  fore- 
saw its  probable  answer.  In  1854,  at  a  celebration  in  com- 
memoration of  the  philosopher  Kant,  he  gave  a  solution  of 
the  problem  under  the  assumption  that  the  sun-  contracts 
in  such  a  way  as  to  remain  always  homogeneous.  With 
our  present  data  regarding  its  rate  of  radiation,  its  volume, 
and  its  mass,  it  is  found  by  the  methods  of  Helmholtz  that, 
under  the  assumption  that  it  is  homogeneous  and  remains 
homogeneous  duririg  its  shrinking,  a  contraction  of  its  radius 
of  120  feet  per  year  would  produce  as  much  heat  as  it  radi- 
ates annually.  This  contraction  is  so  small  that  it  could 
not  be  detected  from  the  distance  of  the  earth  with  our  most 
powerful  telescopes  in  less  than  10,000  years. 

So  far  in  this  discussion  it  has  been  assumed  that  the  sun 
contracts,  and  the  consequences  of  the  contraction  have  been 
deduced.  It  remains  to  consider  the  question  whether  under 
the  conditions  which  prevail  it  actually  does  contract.  The 
reason  it  does  not  at  once  shrink  under  the  mutual  gravita- 
tion of  its  parts  is  that  its  high  temperature  gives  it  a  great 
tendency  to  expand.  As  it  radiates  energy  into  space  its 
temperature  doubtless  falls  a  little;  the  decrease  in  tem- 
perature permits  it  to  contract  a  little ;  the  contraction  pro- 
duces heat  which  momentarily  restores  the  equilibrium; 
and  so  on  in  an  endless  cycle.  This  conclusion  is  certainly 
correct,  as  Ritter  and  Lane  proved  about  1870,  provided 
the  sun  behaves  as  a  monatomic  gaseous  body.  Moreover, 
Lane  established  the  fact,  known  as  Lane's  paradox,  that 
so  long  as  a  purely  gaseous  body  cools  and  contracts,  its 
temperature  rises,  because,  with  decreasing  volume  and 
greater  concentration  of  matter,  the  gravitational  forces 
can  withstand  stronger  expansive  tendencies  due  to  high 
temperature.  If,  with  increasing  concentration,  the  laws  of 
gases  fail  because  the  deep  interior  becomes  liquid  or  soUd, 
the  temperature  might  no  longer  increase. 


358      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  216 

The  question  of  the  variation  in  the  rate  of  radiation  of  a 
contracting  sun  with  increasing  age  is  an  important  one. 
Lane  showed  that,  so  long  as  the  sun  obeys  the  law  of  gases, 
its  temperature  is  inversely  as  its  radius.  By  Stefan's  law 
the  rate  of  radiation  is  proportional  to  the  fourth  power  of 
the  absolute  temperature.  Consequently  the  rate  of  radi- 
ation, per  unit  area,  of  a  contracting  gaseous  sphere  is  in- 
versely as  the  fourth  power  of  its  radius.  But  the  whole 
radiating  surface  is  proportional  to  the  square  of  the  radius. 
Therefore  the  rate  of  radiation  of  the  entire  surface  of  a 
contracting  gaseous  sphere  is  inversely  as  the  square  of  its 
radius.  That  is,  according  to  this  theory,  the  earth  received 
continually  more  and  more  heat  until  the  sun  ceased  to  be 
perfectly  gaseous,  if,  indeed,  it  has  yet  reached  that  stage. 
When  the  sim's  radius  was  twice  as  great  as  it  is  at  present 
it  gave  the  earth  one  fourth  as  much  heat,  and  the  theoretical 
temperature  of  the  earth  (Art.  172)  was  about  200  degrees 
lower  than  at  present. 

217.  Other  Theories  of  the  Sun's  Heat.  —  A  number  of 
other  hjrpotheses  as  to  the  source  of  the  sun's  energy  have 
been  advanced,  but  they  are  all  inadequate.  They  will  be 
enumerated  here  in  order  that  the  reader  may  not  suppose 
that  they  are- important,  and  that  astronomers  have  failed 
to  consider  them. 

The  most  obvious  suggestion  is  that  the  sun  started  hot 
and  is  simply  coohng.  If  it  had  the  very  high  Specific  heat 
of  water,  at  its  present  rate  of  radiation  its  mean  temperature 
would  fall  2.57  degrees  annually.  On  referring  to  its  present 
temperature,  it  is  seen  that  its  radiation  could  not  continue 
more  than  a  few  thousand  years,  and  that  a  few  thousand 
years  ago  its  rate  of  radiation  must  have  been  several  times 
that  at  present.  These  results  are  absurd  and  show  the 
falsity  of  the  suggestion. 

It  is  natural  to  associate  heat  with  something  burning, 
and  one  naturally  inquires  whether  the  heat  of  the  sun 
cannot  be  accounted  for  by  the  combustion  of  the  material 


CH.  XI,  217]  THE   SUN  359 

of  which  it  is  composed.  In  considering  this  hypothesis  the 
first  thing  to  be  noted  is  that  the  same  material  will  burn 
only  once.  It. is  found  from  the  amount  of  heat  produced 
by  coal  that  if  the  sim  were  entirely  made  up  of  the  best 
anthracite  coal  and  oxygen  in  such  proportion  that  when 
the  combustion  was  completed  there  would  be  no  residue 
of  either,  the  heat  generated  would  supply  the  present  rate 
of  .radiation  less  than  1500  years.  If  none  of  the  heat  pro- 
duced by  the  combustion  were  radiated  away,  and  if 
the  specific  heat  of  the  sun  were  unity,  the  temperature  of 
the  sun  would  rise  to  only  about  one  third  of  its  present 
value.  Consequently  this  theory  is  even  less  satisfactory 
than  the  preceding. 

Shortly  after  the  discovery  of  the  law  of  the  conservation 
of  energy  the  large  amount  of  heat  generated  by  the  impact 
of  meteors  was  established.  The  heat  generated  by  a 
meteor  striking  into  the  earth's  atmosphere  at  the  average 
rate  of  25  miles  per  second  is  about  100  times  as  great  as 
would  be  produced  by  its  combustion  if  it  were  oxygen  and 
anthracite  coal.  A  meteor  would  fall  into  the  sun  from 
the  distance  of  the  earth  with  a  velocity  of  about  380  miles 
per  second,  and  since  the  energy  is  proportional  to  the  square 
of  the  velocity,  the  heat  generated  would  be  about  23,000 
times  that  produced  by  the  combustion  of  an  equal  amount 
of  carbon  and  oxygen.  Lord  Kelvin  supposed  that  possibly 
enough  meteors  strike  into  the  sun  to  replenish  the  energy 
it  loses  by  radiation. 

A  complete  answer  to  the  meteoric  theory  of  the  sun's 
heat  is  that  it  requires  an  impossibly  large  total  mass  for 
the  meteors.  They  could  not  possibly  exist  in  sufficient 
numbers  within  the  earth's  orbit;  and,  if  they  came  from 
without,  they  would  strike  the  earth  in  enormously  greater 
numbers  than  are  observed.  In  fact,  computation  shows 
that  if  the  heat  of  the  sun  were  due  to  meteors  coming  into 
it  from  all  directions  and  from  beyond  the  earth's  orbit,  the 
earth  would  receive  -^  as  much  heat  directly  ffom  the 


360      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  217 

meteors  as  it  receives  from  tiie  sun.  This  is  millions  of  times 
more  heat  than  the  earth  receives  from  meteors,  and,  conse- 
quently, the  theory  that  the  sun's  heat  is  maintained  by  the 
impact  of  meteors  is  untenable. 

218.  The  Past  and  the  Future  of  the  Sun  on  the  Basis 
of  the  Contraction  Theory.  —  The  contraction  theory  of 
the  sun's  heat  is  the  only  one  of  those  considered  which 
even  begins  to  satisfy  the  conditions  a  successful  theory  must 
meet.  If  it  is  the  only  important  source  of  the  sun's  heat, 
it  is  possible  to  determine,  at  least  roughly,  how  long  the 
sun  can  have  been  radiating  at  its  present  rate,  and  how 
long  it  can  continue  to  radiate  in  the  future. 

Computation  shows  that  if  the  sun  had  contracted  from 
infinite  expansion,  the  widest  possible  dispersion,  the  total 
amount  of  heat  generated  would  have  been  less  than  20,000,000 
times  the  amount  now  radiated  annually.  If  it  had  con- 
tracted only  from  the  distance  of  the  earth's  orbit,  the  amount 
of  heat  that  would  have  been  generated  would  have  been 
about  one  half  of  one  per  cent  less.  Therefore,  according 
to  the  contraction  theory,  the  earth  can  have  received  heat 
from  the  sun  at  its  present  rate  only  about  20,000,000 
years.  If  the  sun  is  strongly  condensed  at  its  center,  this 
time  limit  should  be  increased  about  5,000,000  years. 

In  the  future,  according  to  this  theory,  the  sun  will  con- 
tract more  and  more  until  it  ceases  to  be  gaseous.  Probably 
by  the  time  its  mean  density  equals  5  its  temperature  will 
begin  to  fall.  A  contraction  to  this  density  will  produce 
enough  heat  to  supply  the  present  rate  of  radiation  only 
10,000,000  years.  Then,  if  the  sun's  contraction  is  the  only 
important  source  of  its  energy,  its  temperature  will  begin  to 
fall,  its  rate  of  radiation  will  diminish,  the  temperature  of 
the  earth  will  gradually  decline,  and  all  life  on  the  earth  will 
eventually  become  extinct.  The  sun,  a  dead  and  invisible  mass, 
will  speed  on  through  space  with  its  retinue  of  lifeless  planets. 

219.  The  Age  of  the  Earth.  —  After  the  development  of 
the  contraction  theory  of  the  sun's  heat,  physicists,  among 


CH.xi,219]  '     THE.  SUN  361 

whom  Lord  Kelvin  was  especially  prominent,  informed  the 
geologists  and  biologists  in  rather  arbitrary  terms  that  the 
earth  was  not  more  than  25,000,000  years  of  age,  and  that 
all  the  great  series  of  changes  with  which  their  sciences 
had  made  them  famiUar  must  have  taken  place  within  this 
time.  But  no  one  science  or  theory  should  be  placed  above 
all  others,  and  other  Unes  of  evidence  as  to  the  age  of  the 
earth  are  .entitled  to  a  full  hearing.  If  they  should  un- 
mistakably agree  that  the  earth  is  much  more  than  25,000,000 
years  of  age,  the  inevitable  conclusion  would  be  that  the 
contraction  theory  is  not  the  whole  truth.  This  is  a  matter 
of  the  greatest  importance,  for  not  only  is  it  at  the  founda- 
tion of  the  interpretation  of  geological  and  biological  evolu- 
tion, but  it  bears  vitally  on  the  question  of  the  age  of  the 
stars  and  on  the  past  and  the  future  of  the  sidereal  universe. 

One  of  the  simplest  methods  employed  by  geologists  for 
determining  the  age  of  the  earth  is  that  of  computing  the 
time  necessary  for  the  oceans  to  acquire  their  sahnity.  The 
rivers  that  flow  into  the  oceans  carry  to  them  various  kinds 
of  salts  in  solution ;  the  water  that  is  evaporated  from  them 
leaves  these  minerals  behind.  Consequently  the  salinity  of 
the  oceans  continually  increases.  It  is  clear  that  it  is 
possible  to  compute  the  age  of  the  oceans  from  the  present 
amount  of  salt  in  them  and  the  rate  at  which  it  is  being 
carried  into  them.  Of  course,  it  is  necessary  to  make  some 
assumptions  regarding  the  rate  at  which  salt  was  carried  to 
the  sea  in  earlier  geological  ages.  The  last  factor  is  some- 
what uncertain,  but  this  method  has  led  to  the  conclusion 
that  the  interval  which  has  elapsed  since  the  oceans  were 
formed  and  salt  began  to  be  carried  down  into  them  is 
more  than  60,000,000  years,  and  that  it  is  probably  from 
90,000,000  to  140,000,000  years. 

Nearly  all  the  rocks  that  are  exposed  on  the  surface  of 
the  earth  are  stratified.  This  means  that,  on  the  whole, 
they  have  been  formed  from  silt  carried  by  the  wind  and 
water  and  deposited  on  the  bottoms  of  lakes  or  oceans. 


362      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  219 

These  stratified  deposits  are  in  many  places  of  enormous 
thickness.  When  it  is  remembered  that  the  present  rocks  are 
usually  not  the  result  of  the  simple  disintegration  and  dep- 
osition of  the  original  earth  material,  but  that  most  of  them 
have  been  repeatedly  broken  up  and  redeposited,  it  is  evi- 
dent that  the  time  required  for  the  great  stratification  which 
is  now  observed  is  enormous.  There  is  obviously  much  chance 
for  divergence  of  views  regarding  the  rates  at  which  these 
processes  have  gone  on,  but  nearly  every  calculation  on  this 
basis  has  led  to  the  conclusion  that  the  time  since  the  dis- 
integration and  stratification  of  the  earth's  rocks  began  is 
at  least  100,000,000  years,  and  most  of  them  have  reached 
much  larger  figures.  The  disintegration  and  total  destruc- 
tion of  mountains  and  plateaus  is  a  closely  related  process 
and  leads  to  the  same  results. 

The  rocks  of  the  earliest  geological  formations  contain 
only  a  few  fossils,  and  they  are  of  primitive  forms  of  life. 
Later  rocks  contain  the  remains  of  higher  forms  of  plants 
and  animals,  until  finally  the  vertebrates  and  the  highest 
types  existing  at  the  present  time  are  found.  Obviously 
an  enormous  interval  of  time  has  been  required  for  all  this 
great  series  of  changes  in  Ufe  forms  to  have  taken  place,  but 
it  is  difiicult  to  make  a  numerical  estimate.  Huxley  gave  the 
question  much  attention  and  thought  a  billion  years  would  be 
necessary  for  the  evolution.  The  recent  discovery  of  muta- 
tion has  shown  that  the  process  of  evolution,  at  least  in  plants, 
may  be  more  rapid  than  he  supposed ;  but,  on  the  whole, 
biologists  feel  that  the  contraction  theory  of  the  sun's  heat 
sets  much  too  restricted  hmits  for  the  age  of  the  earth. 

The  most  recent,  and  possibly  the  best,  method  of  arriv- 
ing at  the  age  of  the  earth  has  followed  the  discovery  of  radio- 
active substances.  Uranium  degenerates  by  a  slow  breaking 
up  of  its  atoms  in  which  radium,  lead,  and  helium  are  evolved. 
From  the  relative  proportions  of  these  products  in  certain 
rocks  it  is  possible  to  compute  the  time  during  which  de- 
generation has  been  going  on'in  them.     This  method  has  led 


CH.  XI,  219]  THE    SUN  363 

to  a  greater  age  for  the  earth  than  any  other.  Strutt,  in  Eng- 
land, Boltwood,  of  Yale,  and  many  others  have  given  this 
method  a  large  amount  of  study,  and  have  obtained  figures 
reaching  up  into  several  hundreds  of  millions  of  years. 
Boltwood,  especially,  has  found  that  the  geologically  older 
rocks  show  greater  antiquity  by  this  method  of  determining 
their  age,  and  he  reaches  the  conclusion  that  some  of  them 
are  nearly  2,000,000,000  years  old. 

It  is  difficult  to  reach  a  positive  conclusion  regarding  the 
age  of  the  earth  from  this  conflicting  evidence.  The  geo- 
logical methods  point  to  an  age  for  the  earth  since  erosion 
began  of  at  least  100,000,000  years.  Geologists  do  not  see 
how  the  facts  in  any  of  their  fines  of  attacking  the  problem 
can  be  brought  into  harmony  with  the  theory  that  the  sun 
has  been  furnishing  fight  and  heat  to  the  earth  for  only 
25,000,000  years.  This  discrepancy  between  their  figures 
and  those  ^ven  by  the  contraction  theory  cannot  be  ig- 
nored, and  therefore  we  are  forced  to  the  conclusion  that 
the  sun  has  other  important  sources  of  heat  energy  besides 
its  contraction.  Aside  from  this,  the  fact  that  a  contract- 
ing gaseous  mass  radiates  inversely  as  the  square  of  its 
radius  gives  a  distribution  of  the  radiation  of  solar  energy 
altogether  at  variance  with  geological  evidence. 

A  possible  source  of  energy  for  the  sun  which  has  not  been 
considered  here  as  yet  is  that  liberated  in  the  degeneration 
of  radioactive  elements.  It  is  not  certain  that  uranium  and 
radium  exist  in  the  sun,  but  helium,  which  is  one  of  the 
products  of  the  disintegration  of  these  elements,  exists  there 
in  abundance;  in  fact,  it  is  called  hefium  because  it  was 
first  discovered  in  the  sun  (Greek,  helios  =  sun),  and  gives 
presumptive  evidence  of  uranium  and  radium  being  there, 
too.  The  disintegration  of  uranium  and  radium  is  accom- 
panied by  the  evolution  of  an  enormous  quantity  of  heat, 
the  energy  Hberated  by  radium  being  about  260,000  times 
that  produced  by  the  combustion  of  an  equal  weight  of  coal 
and  oxygen.     These  results  are  startling,  and  at  first  it 


364      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  219 

seems  that  if  a  small  fraction  of  the  sun  were  radium  or 
uranium,  its  radiation  of  energy  would  be  almost  indefinitely- 
prolonged. 

If  one  part  in  800,000  of  the  sun  were  radium,  heat  would 
be  produced  from  this  source  alone  as  fast  as  it  is  now  being 
radiated,  but  in  less  than  2000  years  half  of  the  radium  would 
be  gone  and  the  production  of  heat  would  correspondingly 
diminish.  Or,  to  go  backward  in  time,  only  2000  years 
ago  the  amount  of  radium  would  have  been  twice  as  great 
as  at  present,  and  the  production  of  heat  would  have  been 
twice  as  rapid.  Since  this  conclusion  is  not  in  harmony 
with  the  facts,  the  hypothesis  that  the  sun's  heat  is  largely 
due  to  the  disintegration  of  radium  is  untenable. 

Now  consider  uranium,  which  degenerates  3,000,000 
times  more  slowly  than  radium.  In  the  case  of  this  element 
the  slowness  of  the  rate  of  degeneration  presents  a  difficulty. 
If  the  sun  were  entirely  uranium,  heat  would  not  be  pro- 
duced more  than  one  third  as  fast  as  it  is  now  being  radi- 
ated. But  in  the  deep  interior  of  the  sun  where  the  tem- 
perature and  pressure  are  inconceivably  high,  the  release  of 
the  subatomic  energies  may  possibly  be  much  more  yapid 
than  under  laboratory  conditions,  and  the  process  may  not 
be  confined  to  the  elements  which  are  radioactive  at  the  sur- 
face of  the  earth.  There  is  no  laboratory  experience  to  sup- 
port this  suggestion  because  within  the  range  of  experiment 
the  rates  of  the  radioactive  processes  have  been  found  to 
be  independent  of  temperature  and  other  physical  con- 
ditions. But,  if  there  is  something  in  the  suggestion,  and 
especially  if  under  the  conditions  prevailing  in  the  sun  the 
subatomic  energies  of  all  elements  are  released,  the  amount 
of  energy  may  be  sufficient  for  hundreds  and  even  thousands 
of  miUions  of  years.  But  at  once  the  question  regarding  the 
origin  of  the  subatomic  energies  arises,  and,  at  present,  there 
is  no  answer  to  it. 


CH.  XI,  220]  THE^  SUN  365 

XV.     QUESTIONS 

1.  How  many  horse  power  of  energy  per  inhabitant  is  received 
by  the  earth  from  the  sun  ? 

2.  What  is  the  average  amount  of  energy  per  square  yard  re- 
ceived by  the  whole  earth  from  the  sun  ? 

3.  Does  the  energy  which  is  manifested  in  the  tides  come  from 
the  sun?     What  becomes  of  the  energy  in  the  tides? 

4.  What  becomes  of  that  part  of  the  sun's  energy  which  is 
absorbed  by  the  earth's  atmosphere?  ' 

5.  If  the  earth's  atmosphere  absorbs  35  per  cent  of  the  energy 
which  comes  to  it  from  the  sun,  how  can  the  atmosphere  cause  the 
temperature  of  the  earth's  surface  to  be  higher  than  it  would  other- 
wise be  ? 

6.  Show  from  the  rate  at  which  the  earth  receives  energy  from 
the  sun,  the  size  of  the  sun,  and  the  earth's  distance  from  the  sun, 
that  the  sun  radiates  70,000  horse  power  of  energy  per  square 
yard. 

7.  Taking  the  earth's  mean  temperature  as  60°  F.  and  the  rates 
of  radiation  of  the  earth  (see  question  2)  and  of  the  sun,  compute 
the  temperature  of  the  sun  on  the  basis  of  Stefan's  law. 

8.  All  scientists  agree  that  the'  earth  is  more  than  5,000,000 
years  old.  On  the  hypothesis  that  ^he  contraction  of  the  sun  is  its 
only  source  of  heat,  and  that  during  the  last  5,000,000  years 
it  has  radiated  at  its  present  rate,  what  were  its  radius  and  density 
at  the  beginning  of  this  period?  On  the  basis  of  Lane's  law,  what 
was  its  temperature?  On  the  basis  of  Stefan's  law,  what  was  its 
rate  of  radiation  per  unit  area  and  as  a  whole  ?  On  the  basis  of  the 
method  of  Art.  172,  what  was  the  mean  temperature  of  the  earth? 

II.   Spectrum  Analysis 

220.  The  Nature  of  Light.  —  In  order  to  comprehend 
the  principles  of  spectrum  analysis  it  is  necessary  to  under- 
stand the  nature  of  light.  A  profound  study  of  the  fun- 
damental properties  of  light  was  begun  by  Newton,  but, 
unfortunately,  some  of  his  basal  conclusions  were  quite 
erroneous.  Thomas  Young  (1773-1829)  laid  the  foundation 
of  the  modern  undulatory  theory  of  light.  That  is,  he 
established  the  fact  that  light  consists  of  waves  in  an  all-per- 
vading medium  known  as  the  ether,  by  showing  that  when 


366      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  szO 

two  similar  rays  of  light  meet  they  destroy  each  other  where 
their  phases  are  different,  and  add  where  their  phases  are 
the  same.  These  phenomena,  which  are  analogous  to  those 
exhibited  by  waves  in  water,  would  not  be  observed  if 
Newton's  idea  were  correct  that  hght  consisted  of  minute 
particles  shot  out  from  a  radiating  body. 

Physical  experiments  prove  that  hght  waves  in  the  ether 
are  at  right  angles  to  the  Hne  of  their  propagation,  like  the 
up-and-down  waves  which  travel  along  a  steel  beam  when 
it  is  struck  with  a  hammer,  or  the  torsional  waves  that  are 
transmitted  along  a  sohd  elastic  body  when  one  of  its  ends 
is  suddenly  twisted.  In  an  ordinary  beam  of  light  the 
vibrations  are  in  every  direction  perpendicular  to  the  hne 
of  propagation.  If  the  vibrations  in  one  direction  are  de- 
stroyed while  those  at  right  angles  to  it  remain,  the  hght 
is  said  to  be  polarized.  Many  substances  have  the  property 
of  polarizing  light  which  passes  through  them. 

The  distance  from  one  wave  to  the  next  for  red  hght  is 
about  4  0,0  0  0  of  3,n  inch,  and  for  violet  light  about  ^p.ooo  of  an 
inch.  There  are  vibrations  both  of  smaller  and  greater  wave 
lengths.  The  range  beyond  the  violet '  is  not  very  great, 
for,  even  though  very  short  waves  are  emitted  by  a  body, 
they  are  absorbed  and  scattered  by  the  earth's  atmosphere 
before  'reaching  the  observer ;  but  there  is  no  hmit  in  the 
other  direction  to  the  lengths  of  rays.  Langley  explored  the 
so-called  heat  rays  of  the  sun  with  his  bolometer  far  be- 
yond those  which  are  Adsible  to  the  human  eye.  The  waves 
used  in  wireless  telegraphy,  which  differ  from  light  waves 
only  in  their  length,  are  often  hundreds  of  yards  long. 

221.  On  the  Production  of  Light.  —  A  definite  concep- 
tion of  the  way  in  which  matter  emits  radiant  energy  is 
important  for  an  understanding  of  the  principles  of  spec- 
trum analysis,  but,  unfortunately,  the  fundamental  proper- 
ties of  matter  are  involved,  and  physicists  are  not  yet  in  agree- 
ment on  the  subject.  However,  the  theory  that  radiant 
•  Excepting  the  so-called  X-rays,  which  are  much  shorter. 


CH.  XI,  221]\  THE    SUN  367 

energy  is  due  to  accelerated  electrons  is  in  good  standing  and 
gives  a  correct  representation  of  the  principal  facts. 

The  molecules  of  which  substances  are  composed  are 
themselves  made  up  of  atoms..  The  atoms  were  generally 
supposed  to  be  indivisible  until  the  year  1895,  when  the 
cathode  and  X-rays  prepared  the  way  for  the  recent  dis- 
coveries in  radioactivity  and  subatomic  units.  In  con- 
nection with  these  discoveries  it  was  found  that  the  atoms 
are  made  up  of  numerous  still  smaller  particles,  called  elec- 
trons or  corpuscles.  An  atom,  according  to  the  hypothesis 
of  Rutherford,  is  composed  of  a  small  central  nucleus,  carry- 
ing a  positive  charge  of  electricity,  and  one  or  more  rings 


Fig.  137.  —  Model  of  atom,  non-radiating  at  left  and  radiating  at  right. 

of  electrons  carrying  (or  perhaps  consisting  of)  negative 
charges  of  electricity,  which  revolve  around  the  positive 
nucleus  at  great  speed.  Under  ordinary  circumstances  the 
electrons  revolve  in  circular  paths  with  uniform  speed,  all 
those  'of  a  given  ring  travehng  in  the  same  circle.  Under 
these  circumstances,  represented  in  the  left  of  Fig.  137, 
the  atom  is  not  radiating. 

When  a  body  is  highly  heated  the  molecules  and  atoms 
of  which  it  is  composed  are  in  very  rapid  motion  and  jos- 
tle against  one  another  with  great  frequency.  These  im- 
pacts disturb  the  motions  of  the  electrons  and  cause  them  to 
describe  wavy  paths  in  and  out  across  the  circles  in  which 
they   ordinarily   move.     This   condition   is   shown   at   the 


368      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  221 

right  in  Fig.  137.  These  small  vibrations,  which  are 
periodic  in  character,  produce  light  waves  in  the  ether; 
and  Hght  waves  are  also  produced  by  the  impacts  themselves, 
but  they  are  not  periodic. 

The  character  of  the  motions  of  the  corpuscles  can  be 
understood  by  considering  a  bell.  Suppose  it  is  suspended 
by  a  twisted  cord  which  is  rapidly  untwisting.  A  ring  of 
particles  around  the  bell  corresponds  to  a  ring  of  corpuscles 
in  an  atom.  If  the  bell  is  simply  rotating,  it  gives  out  no 
sound.  Suppose  it  strikes  something.  The  particles  of 
which  it  is  composed  vibrate  rapidly  in  and  out ;  this,  com- 
bined with  its  rotation,  causes  them  to  describe  wavy  paths 
across  their  former  circular  orbits.  These  waves  produce 
the  sound.  Of  course,  it  is  not  necessary  that  the  bell  should 
be  rotating  in  order  to  produce  sound,  and  in  this  respect 
the  analogy  is  imperfect. 

The  frequency  of  the  vibrations  of  a  corpuscle  in  an  atom 
is  astounding.  The  length  of  a  light  wave  of  yellow  light 
is  in  round  numbers  50,000  of  an  inch.  In  a  second  of  time 
enough  waves  are  emitted  to  make  a  line  of  them  186,000 
miles  along.  Therefore,  the  number  of  oscillations  per 
second  of  the  corpuscles  in  an  atom  is  in  round  numbers 
600,000,000,000,000. 

It  has  often  been  suggested  that  the  atoms  of  all  the 
chemical  elements  are  made  out  of  exactly  the  same  kind  of 
electrons.  Certainly  there  is  as  yet  no  evidence  to  the  con- 
trary. If  the  electrons  are  not  composite  structures  them- 
selves, the  idea  is  reasonable  enough ;  but  if  they  are  made 
up  of  still  smaller  units,  the  hypothesis  seems  improbable. 

The  dynamics  of  an  atom,  according  to  the  corpuscular 
theory,  is  of  much  interest.  The  positive  nucleus  attracts 
the  revolving  negative  corpuscles.  They  are  kept  from  fall- 
ing in  on  the  nuclelis  both  by  the  centrifugal  force  due  to 
their  rapid  revolution,  and  also  by  their  mutual  repulsions 
which  result  from  their  being  similarly  electrified.  If  the 
number  of  corpuscles  in  a  ring  is  small,  the  atom  is  stable. 


CH.  XI,  222]  THE    SUN  369 

With  an  increasing  number  of  corpuscles  the  stabiHty  of  the 
atom  diminishes.  Finally,  the  atom  is  stable  only  if  the 
corpuscles  revolve  in  two  or  more  rings.  The  regions  of 
instability  which  separate  atoms  having  a  certain  number 
of  rings  from  those  having  other  numbers  possibly  give  a 
clue  to  the  celebrated  periodic  law  of  the  chemical  elements 
discovered  by  Mendel6eff. 

222.  Spectroscopes  and  the  Spectrum.  —  The  energy 
which  a  body  radiates  is  completely  characterized  by  the 
wave  lengths  which  it  includes  and  their  respective  inten- 
sities. The  spectroscope  is  an  instrument  which  enables  us 
to  analyze  light  into  its  parts  of  different  wave  lengths,  and 
to  study  each  one -separately. 

There  are  three  principal  types  of  spectroscopes.  In  the 
first  and  oldest  type  the  light  passes  through  one  or  more 
prisms ;  in  the  second,  perfected  by  Rowland  and  Michelson, 
the  light  is  reflected  from  a  surface  on  which  are  ruled 
many  parallel  equidistant  lines ;  and  in  the  third,  invented 
by  Michelson,  the  light  passes  through  a  pile  of  equally 
thick  plane  pieces  of  glass  piled  up  like  a  stairway.  The 
first  type  is  most  advantageous  when  the  source  of  light  is 
faint,  Uke  a  small  star,  comet,  or  nebula.  Its  chief  fault  is 
that  the  scale  of  the  spectrum  is  not  the  same  in  all  parts. 
The  second  type  is  advantageous  for  bright  sources  of  hght 
like  the  sun  or  the  electric  arc  in  the  laboratory.  It  gives 
the  same  scale  for  all  parts  of  the  spectrum,  but  uses  only 
a  small  part  of  the  incident  light.  The  third  type,  known 
as  the  echelon,  gives  high  dispersion  without  great  loss  of 
light.  Only  the  first  type,  which  is  most  used  in  astronomy, 
will  be  more  fully  described  here. 

The  basis  of  the  prism  spectroscope  is  the  refraction  and 
the  dispersion  of  light  when  it  passes  through  a  prism.  Let 
L,  Fig.  138,  represent  a  beam  of  white  light  which  passes 
through  the  prism  P.  As  it  enters  at  A  from  a  rarer  to  a 
denser  medium,  it  is  bent  toward  the  perpendicular  to  the 
surface;  and  as  it  emerges  at  B  from  &■  denser  to  a  rarer 
2b 


370      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  222 

medium,  it  is  bent  from  the  perpendicular  to  the  surface. 
This  change  in  the  direction  of  the  beam  of  Hght  is  its 
refraction. 

Not  only  is  the  beam  of  light  refracted,  but  it  is  also  spread 
out  into  its  colors.  As  it  enters  the  prism  the  violet  light 
is  refracted  the  most  and  the  red  the  least,  and  the  same  thing 
is  true  when  it  emerges.  Consequently,  instead  of  a  beam 
of  white  light  falling  on  the  screen  S  there  is  found  a  band  of 
colors  which,  in  order  from  the  most  refracted  to  the  least 

refracted,  are  violet, 
^^-''-  indigo,  blue,  green, 
yellow,  orange,  and 
red.  This  separation 
of  light  into  its  colors 
is  called  dispersion. 

In  the  diagram  only 
the  visible  part  of  the 

Fig.   138.  —  Refraction  and  dispersion  of  light    spectrum  is  indicated. 

y  a  pnsm.  Beyond   the   red    are 

the  infra-red,  or  heat,  rays  I-R,  and  beyond  the  violet  are 
the  ultra-violet  rays  U-V.  The  colors  are  not  separated  by 
sharp  boundaries,  but  shade  from  one  to  another  by  insensi- 
ble gradations.  The  ultra-violet  part  of  the  spectrum  is 
several  times  as  long  as  the  visible  part,  and  the  infra-red 
part  is  several  times  as  long  as  the  ultra-violet  part. 

While  Fig.  138  shows  exactly  the  way  in  which  a  spec- 
trum might  be  formed,  it  would  be  too  faint  to  be  of  any 
value  in  practice.  In  order  to  obtain  a  bright  spectrum  the 
apparatus  is  arranged  as  sketched  in  Fig.  139,  though  in 
practice  several  prisms,  one  after  the  other,  are  often  em- 
ployed. The  rays  which  pass  through  the  screen  at  0  are 
made  parallel  by  the  lens  Lj.  They  strike  the  prism  P  in 
parallel  lines,  and  those  of  a  given  color  continue  through  P 
and  to  the  lens  L^  in  parallel  lines  (the  dispersion  is  not  indi- 
cated in  the  diagram).  The  lens  La  brings  the  rays  to  a 
focus  at  F,  and  the  eyepiece  E  sends  all  those  of  each  color 


CH.  XI,  223]  THE    SUN  371 

out  in  a  small  bundle  of  parallel  lines  (only  one  color  is  repre- 
sented in  the  diagram).  The  eye  is  placed  just  to  the  right 
of  E,  and  all  the  parallel  rays  of  each  bundle  are  brought  to 
a  focus  at  a  point  on  the  retina.  In  this  way  many  rays  of 
each  color  are  brought  to  a  focus  at  the  same  place  in  the 
observer's  eye. 

While  strictly  white  light  gives. all  colors,  it  is  not  neces- 
sary that  a  luminous  body  should  emit  all  kinds  of  light,  or 
that  all  colors  emitted  should  be  given  out  in  equal  intensity. 
In  fact,  it  is  well  known  that  if  a  body  is  simply  warm  but 
not  self-luminous,  it  gives  out  in  sensible  quantities  only 


Fig,   139.  —  A  spectroscope  having  only  one  prism. 

infra-red  rays.  If  it  is  extremely  hot,  it  may  radiate  mostly 
ultra-violet  rays. 

223.  The  First  Law  of  Spectrum  Analysis.  —  The  first 
theoretical  discussion  of  the  principles  of  spectrum  analysis 
which  reached  approximately  correct  conclusions  was  made 
by  Angstrom  in  1853.  The  work  of  Bunsen,  and  especially 
of  Kirchhoff  in  1859,  put  the  subject  on  essentially  its  present 
basis.  The  laws  of  spectrum  analysis  as  formulated  here 
are  consequences  of  a  general  law  due  to  Kirchhoff,  and  of 
certain  experimental  facts.  After  they  have  been  stated, 
they  will  be  seen  to  be  simple  consequences  of  the  mode  of 
production  of  radiant  energy. 

The  first  law  of  spectrum  analysis  is :  A  radiating  solid, 
liquid,  or  gas  under  high  pressure  gives  a  continuous  spectrum 
whose  position  of  maximum  intensity  depends  upon  the  tem- 
perature of  the  source;   and  conversely,  if  a  spectrum  is  con- 


372      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  223 

tinuous,  the  source  of  light  is  a  solid,  liquid,  or  gas  under  high 
pressure,  and  the  position  of  radiation  of  maximum  intensity 
determines  the  temperature  of  the  source. 

This  law  means,  in  the  first  place,  that  a  radiating  solid, 
liquid,  or  gas  under  high  pressure  gives  out  light,  or  more 
generally  radiant  energy,  of  all  wave  lengths;  and,  in  the 
second  place,  the  wave  length  at  which  the  radiation  is  most 
intense  depends  upon  the  temperature  of  the  source.  It  is 
clear  from  the  way  in  which  light  is  produced  that  the  first 
part  of  the  law  should  be  true.  When  a  body  is  in  a  sohd 
or  liquid  state,  or  when  it  is  a  gas  under  high  pressure,  the 
molecules  are  so  close  together  that  they  continually  inter- 
fere with  one  another.  Under  these  circumstances  the  os- 
cillations of  the  corpuscles  cannot  take  place  in  their  natural 
periods,  but  they  are  altered  in  all  possible  manners.  This 
results  in  vibrations  of  all  periods,  and  therefore  the  spectra 
are  continuous. 

The  way  in  which  the  wave  length  of  maximum  radiation 
depends  upon  the  temperature  is  given  by  Wien's  law  ^  — 

_  0.2076 

where  A.  is  the  wave  length  in  inches  and  T  is  the  absolute 
temperature  on  the  Fahrenheit  scale.  For  example,  if  the 
temperature  of  the  sun  is  10,000°,  its  wave  length  of  maxi- 
mum radiation  is  about  50,000  of  an  inch. 

224.  The  Second  Law  of  Spectrum  Analysis.  —  The 
second  law  of  spectrum  analysis  is :  A  radiating  gas  under 
low  pressure  gives  a  spectrum  which  consists  of  bright  lines 
whose  relations  to  one  another  and  whose  positions  in  the 
spectrum'^  depend  upon  the  nature  of  the  gas  (and  in  some 

'Experiments  show  that  this  law  does  not  give  good  results  for  low 
temperatures,  but  the  applications  in  astronomy  are  to  high  temperatures. 

^The  positions  of  lines  in  a  spectrum  determine,  of  course,  their  relations 
to  one  another ;  but  in  practice  the  lines  of  an  element  aire  usually  identified 
by  their  relations  to  one  another,  just  as  a  constellation  is  recognized  by  the 
relative  positions  of  its  stars. 


CH.  XI,  224] 


THE    SUN 


373 


cases  to  some  extent  upon  its  temperature,  density,  electrical 
and  magnetic  condition);  and  conversely,  if  a  spectrum  con- 
sists of  bright  lines,  then  the  source  is  a  radiating  gas  (or 
gases)  under  low  pressure,  and  the  composition  of  the  gas  (or 
gases)  can  be  determined  from  the  relations  of  the  lines  to  one 
another  and  from  their  positions  in  the  spectrum. 

When  molecules  are  free  from  all  restraints  the  oscil- 
lations of  their  electrons  take  place  in  fixed  periods  which 
depend  upon  the  internal  forces  involved,  just  as  free  bells 
of  given  structure  vibrate  in  definite  ways  and  give  forth 
sounds  of  definite  pitch.  Consequently,  free  radiating  mole- 
cules emit  light  of  one  or  more  definite  wave  lengths  de- 
pending on  the  structure  of  the  molecules,  and  there  are 


Fig.  140.  —  A  bright-line  speotrum  above  and  a  reversed  spectrum  below. 

bright  lines  at  corresponding  places  in  the  spectrum  and  no 
hght  whatever  at  other  places.  A  bright-line  spectrum  is 
shown  in  the  top  part  of  Fig.  140.  Some  elements  give 
only  a  few  lines  and  others  a  great  many.  For  example, 
sodium  has  but  two  Knes,  both  in  the  yellow,  and  iron  more 
than  2000  lines.  It  is  needless  to  say  that  all  these  facts 
are  established  by  laboratory  experiments. 

It  may  be  objected  that  in  a  gas,  even  under  low  pressure, 
the  molecules  are  not  free  from  outside  interference,  for  they 
colhde  with  one  another  many  millions  of  t^mes  per  second. 
But  the  intervals  during  which  they  are  in  collision  are  very 
short  compared  with  the  intervals  between  collisions.  Con- 
sequently, while  there  will  be  some  light  of  all  wave  lengths, 
it  will  be  inappreciable  compared  to  that  which  is  character- 
istic of  the  radiating  gas,  and  the  spectrum  will  seem  to  con- 


374      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  224 

sist  of  bright  lines  of  various  colors  on  a  perfectly  black 
background. 

225.  The  Third  Law  of  Spectrum  Analysis.  —  The  third 
law  of  spectrum  analysis  is :  //  light  from  a  solid,  liquid,  or 
gas  under  great  pressure  passes  through  a  cooler  gas  (or  gases), 
then  the  result  is  a  bright  spectrum  which  is  continuous  except 
where  it  is  crossed  by  dark  lines,  and  the  dark  lines  have  the 
positions  which  would  be  occupied  by  bright  lines  if  the 
intervening  cooler  gas  were  the  source  of  light;  and  con- 
versely, if  a  bright  spectrum  is  continuous  except  where  it  is 
crossed  by  dark  lines,  then  the  source  of  light  is  a  solid,  liquid, 
or  gas  under  great  pressure,  and  the  light  has  passed  through 
a  cooler  intervening  gas  (or  gases)  whose  constitution  can  be 
determined  from  the  relations  of  the  dark  lines  to  one  another 
and  from  their  positions  in  the  spectrum. 

In  a  word,  a  cool  gas  absorbs  the  same  kinds  of  rays  it 
would  give  out  if  it  were  incandescent,  and  no  others.  Simi- 
larly, a  musical  instrument  absorbs  tones  of  the  same  pitch 
as  those  which  it  can  produce.  For  example,  if  the  key  for 
middle  C  on  a  piano  is  held  down  and  this  tone  is  produced 
near  by,  the  piano  will  respond  with  the  same  tone ;  but  if 
D  is  produced,  the  piano  will  give  no  response.  This  phe- 
nomenon occurs  in  many  branches  of  physics  and  is  very 
important.  For  example,  it  is  at  the  basis  of  wireless  teleg- 
raphy. The  receiving  instrument  and  the  sending  instru- 
ment are  tuned  together,  and  only  in  this  way  do  the  effects 
of  the  feeble  waves  which  reach  to  great  distances  become 
sensible.  The  fact  that  the  sending  and  receiving  instru- 
ments must  be  tuned  the  same  explains  how  it  is  that  many 
different  wireless  instruments  can  be  working  at  the  same 
time  without  sensible  interference. 

When  the  intervening  cooler  gas  absorbs  certain  parts  of 
the  energy  which  passes  through  it,  it  becomes  heated  and 
its  rate  of  radiation  is  increased.  It  might  be  supposed  that 
this  new  radiation  would  make  up  for  the  energy  which  has 
been  absorbed.     That  which  has  been  absorbed  and  that 


CH.  XI,  226]  THE    SUN  375 

which  is  radiated  are,  indeed,  exactly  equal,  but  the  radi- 
ated energy  is  sent  out  in  every  direction  and  not  alone  in 
the  direction  of  the  original  light  passing  through  the  gas. 
That  is,  certain  parts  of  the  original  energy  are  taken  out 
and  scattered  in  every  direction.  Therefore,  in  a  spectrum 
crossed  by  dark  Unes  the  dark  lines  are  not  absolutely  black, 
but  only  black  relatively  to  the  remainder  of  the  spectrum. 
A  spectrum  of  this  sort  is  called  an  absorption,  or  dark -line, 
or  reversed  spectrum.  The  reverse  of  the  bright-hne  spec- 
trum given  in  the  top  of  Fig.  140  is  shown  in  the  bottom 
part  of  the  figure. 

226.  The  Fourth  Law  of  Spectrurm  Analysis.  —  The  fourth 
law  of  spectrum  analysis  was  first  discovered  by  Doppler 
and  was  experimentally  established  by  Fizeau.  It  is  com- 
monly called  the  Doppler  principle,  or  the  Doppler-Fizeau 
law.  It  is  :  //  the  source  (radiating  gas  in  the  case  of  a  spec- 
trum of  bright  lines,  and  an  intervening  cooler  gas  in  case  of 
an  absorption  spectrum)  and  receiver  are  relatively  approach- 
ing toward,  or  receding  from,  each  other,  then  the  lines  of  the 
spectrum  are  displaced  respectively  in  the  direction  of  the 
violet  or  the  red  by  an  amount  which  is  proportional  to  the 
relative  speed  of  approach  or  recession;  and  conversely,  if  the 
lines  of  a  spectrum  are  displaced  toward  the  violet  or  the  red, 
the  source  and  receiver  are  respectively  approaching  toward, 
or  receding  from,  each  other,  and  the  relative  speed  of  approach 
or  recession  can  be  determined  from  the  amount  of  the  dis- 
placement. 

The  explanation  of  the  shift  of  the  hues  of  the  spectrum 
when  there  is  relative  motion  of  the  source  and  the  re- 
ceiver is  very  simple.  If  the  source  is  stationary,  it  sends 
out  wave  after  wave  separated  by  a  given  interval ;  if  it  is 
moving  toward  the  receiver,  it  follows  up  the  waves  which  it 
emits  and  the  intervals  between  them  are  diminished.  That 
is,  the  wave  lengths  have  become  shorter,  which  is  only  an- 
other way  of  stating  that  the  corresponding  spectral  lines 
have  been  shifted  toward  the  violet.    Of  course,  for  motion 


376      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  226 

in  the  opposite  direction  the  spectral  Unes  are  shifted  toward 
the  red. 

If  the  receiver  moves  toward  the  source,  he  receives  not 
only  the  waves  which  would  reach  him  if  he  were 
stationary,  but  also  those  which  he  meets  as  a  conse- 
quence of  his  motion.  The  distances  between  the  waves 
are  diminished  and  the  spectral  lines  are  shifted  toward 
the  violet.  Motion  in  the  opposite  direction  produces  the 
opposite  results. 

The  formula  for  the  shift  in  the  spectral  lines  is 

where  A\  is  the  amount  of  the  shift,  X  is  the  wave  length 
of  the  line  in  question,  v  the  relative  velocity  of  the  source 
and  receiver,  and  V  the  velocity  of  light.  Suppose  v  is  18.6 
miles  per  second ;  then,  since  V  is  186,000  miles  per  second 
and  the  greatest  wave  length  in  the  visible  spectrum  is 
nearly  twice  that  of  the  shortest,  the  displacement  is  about 
10.000  of  the  distance  between  the  ends  of  the  visible  spec- 
trum. It  follows  that  for  the  velocities  with  which  the 
planets  move  the  displacements  of  the  spectral  lines  are 
very  small,  and  that  refined  means  must  be  employed  in 
order  to  determine  them  accurately.  The  usual  method  is 
to  photograph  the  spectrum  of  the  distant  object  and  at 
the  same  time  to  send  through  the  spectroscope  beside  it 
the  hght  from  some  suitable  laboratory  source.  The  lines, 
of  the  latter  will  of  course  have  their  normal  positions. 
The  displacements  of  the  hnes  of  the  celestial  object  with 
respect  to  them  are  measured  with  the  aid  of  a  micro- 
scope. 

When  the  spectral  hnes  of  an  object  are  well  defined,  dis- 
placement results  of  astonishing  precision  can  be  obtained. 
In  the  case  of  stars  of  certain  types  the  relative  velocities 
toward  or  from  the  earth,  called  radial  velocities,  can  be  de- 
termined to  within  one  tenth  of  a  mile  per  second. 


CH.  XI,  227]  THE    SUN  377 

XVI.    QUESTIONS. 

1.  What  problems  can  be  solved  approximately  for  the  sun  and 
stars  by  the  first  principle  of  spectrum  analysis  ? 

2.  What  would  be  the  character  of  the  spectrum  of  moonlight  ? 

3.  Comets  have  continuous  bright  spectra  crossed  by  still  brighter 
lines  ;  what  interpretation  is  to  be  made  of  these  facts,  remembering 
that  comets  shine  partly  by  reflected  light  ? 

4.  The  spectra  of  Uranus  and  Neptune  contain  dark  lines  and 
bands  of  great  intensity  at  the  positions  of  the  less  intense  hydrogen 
lines  of  the  solar  spectrum  ;  what  interpretation  is  to  be  placed  on 
these  phenomena  ? 

5.  Can  the  motion  of  the  earth  With  respect  to  the  sun  and  moon 
be  determined  by  spectroscopic  means  ?  The  motion  of  the  earth 
with  respect  to  the  planets? 

6.  If  an  observer  were  approaching  a  deep  red  star  with  the  veloc- 
ity of  light,  what  color  would  the  star  appear  to  have  ?  If  he  were 
receding  with  the  velocity  of  light  ? 

7.  What  effect  would  the  rapid  rotation  of  a  star  have  on  its  spec- 
tral lines  ? 

8.  Suppose  an  observer  examines  the  spectra  of  the  eastern  and 
western  Umbs  of  the  sun ;  how  would  the  spectral  lines  be  related  ? 
Could  they  be  distinguished  from  lines  due  to  absorption  by  the 
earth's  atmosphere '! 

III.   The  Constitution  of  the  Sun 

227.  Outline  of  the  Sun's  Constitution.  —  The  apparent 
surface  of  the  sun  is  called  the  -photosphere  (hght  sphere). 
It  has  the  appearance  of  being  rather  sharply  defined,  Fig. 
141,  and  it  is  the  boundary  used  to  define  the  size  of  the  sun, 
but  the  sun  is  disturbed  by  such  violent  vertical  motions 
that  it  is  probably  very  broken  in  outline.  At  the  dis- 
tance of  the  sun  from  the  earth  an  object  500  miles  across 
subtends  an  angle  of  only  one  second  of  arc,  and,  therefore, 
irregularities  in  the  photosphere  would  not  be  visible  unless 
they  amounted  to  several  hundred  miles.  The  part  of  the 
sun  interior  to  the  photosphere  is  always  invisible. 

Above  the  photosphere  hes  a  sheet  of  gas,  probably  from 
500  to  1000  miles  thick,  which  is  called  the  reversing  layer 
because,  as  will  be  seen  (Art.  233),  it  produces  a  reversed, 


378      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  227 

or  absorption,  spectrum.     It  contains  many  terrestrial  sub- 
stances, such  as  calcium  and  iron,  in  a  vaporous  state. 

Outside  of  the  reversing  layer  is  another  layer  of  gas, 
from  5000  to  10,000  miles  deep,  called  the  chromosphere 


Fig.   141.  —  The  Sun.     Photographed  by  Fox  with  the  Jfi-inch  telescope  of  the 
Yerkes  Observatory. 

(color  sphere).     At  the  time  of  a  total  eclipse  of  the  sun  it 
is  seen  as  a  brilliant  scarlet  fringe  whose  outer  surface  seems 


CH.  XI,  228]  THE    SUN  379 

to  be  covered  with  leaping  flames.  There  are  often  eruptions, 
called  prominences,  which  break  up  into  it  and  ascend  to 
great  heights. 

The  outermost  portion  of  the  sun  is  the  corona  (crown), 
a  halo  of  pearly  light  which  is  so  much  fainter  than  the  il- 
lumination of  the  earth's  atmosphere  that  it  can  be  seen  only 
at  the  time  of  a  total  solar  eclipse.  It  is  irregular  in  form  and 
gradually  fades  out  into  the  blackness  of  the  sky  at  the  dis- 
tance of  from  1,000,000  to  3,000,000  miles  from  the  surface 
of  the  sun. 

Figure  142  shows  an  ideal  section  through  the  sun.  The 
upper  surface  of  the  invisible  interior  is  the  photosphere, 


/       / 


*    f.  «i,ji 


1 1 


I"    I     ■        '  T  >    i  I 


1 


"  Fig.  142.  —  Ideal  section  of  the  sun. 

R  is  the  reversing  layer,  »S  is  a  spot,  K  is  the  chromosphere, 
P  is  a  prominence,  and  C  is  the  corona. 

228.  The  Photosphere.  —  When  the  sun  is  examined 
through  a  good  telescope  it  presents  a  finely  mottled  ap- 
pearance instead  of  the  uniform  luster  which  might  be  ex- 
pected. The  brighter  parts  are  intensely  luminous  nodules, 
somewhat  irregular  in  form,  500  or  600  miles  across.  These 
"  rice  grains,"  as  they  are  sometimes  called,  have  been  re- 
solved into  smaller  elements  having  a  diameter  of  not  over 
100  miles ;  and  although  all  these  granules  together  do  not 
constitute  over  one  fifth  of  the  sun's  surface,  yet,  according 


380      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  228 


to  Langley's  estimates,  they  radiate  about  three  fourths  of 
the  hght.  A  small  portion  of  the  sun's  surface  highly 
magnified  is  shown  in  Fig.  143. 

The  photosphere  of  the  sun  gives  a  continuous  spectnmi. 
Therefore,  according  to  the  first  law  of  spectrum  analysis, 
it  is  a  solid,  hquid,  or  gas  under  great  pressure.  Since  the 
photosphere  is  not  transparent  there  is  a  strong  inchnation 
to  infer  that  it  is  hquid,  or  at  least  consists  of  clouds  of 
liquid  particles  (carbon,  iron,  calcium,  etc.)  floating  in  a 

vapor  of  similar  substances. 
But  the  temperature  of  the 
sun  is  so  high  that  this 
conclusion  is  not  certain. 

In  considering  the  sun  it 
must  be  remembered  that 
its  surface  gravity  is  nearly 
28  times  that  of  the  earth, 
and  that  the  pressure  under 
equal  masses  of  atmosphere 
is  correspondingly  greater. 
Hence,  it  is  not  unreasona- 
ble to  suppose  that  the 
pressure  down  under  the 
corona,  chromosphere,  and 
reversing  layer  is  great  enough  to  produce  a  continuous 
spectrum.  The  conclusion  that  the  photosphere  is  almost 
entirely,  if  not  altogether,  gaseous  is  supported  by  the  fact 
that  the  cooler,  overlying  reversing  layer  is  gaseous  and 
contains  some  of  the  most  refractory  known  substances. 
The  "  rice-grain  "  structure  of  the  photosphere  is  explained 
by  Abbott  as  being  due  to  relative  motions  of  layers  at 
different  levels  analogous  to  those  which  produce  a  mackerel 
sky  in  the  earth's  atmosphere.  He  supposes  that  the  dark 
places  between  the  "  rice-grains  "  correspond  to  those  places 
where  clouds  form  in  our  own  atmosphere,  and  that  they 
are  regions  where  the  tpmperature  has  fallen  somewhat 


Fig.   143.  —  Small  portion  of  the  sun's 
surface,  highly  magnified. 


CH.  XI,  229]  THE    SUN  381 

below  that  of  the  remainder  of  the  photosphere.  There  are 
other  astronomers,  however,  who  beheve  that  the  bright  nod- 
ules are  the  summits  of  ascending  convection  currents,  which, 
by  expansion  and  cooling,  are  reduced  to  the  state  where  the 
most  refractory  substances  partially  condense  and  radiate 
most  briUiantly,  while  the  darker  spaces  between  are  where 
the  cooler  currents  descend. 

The  photosphere  is  the  region  from  which  the  sun  loses 
energy  by  radiation.  This  energy  must  be  suppUed  from 
the  interior.  There  are  three  processes  by  which  heat  may 
be  transferred  from  one  position  to  another,  viz.,  by  conduc- 
tion, by  convection,  and  by  radiation.  Conduction  is  en- 
tirely too  slow  to  be  quantitatively  adequate  for  bringing 
heat  to  the  surface  of  the  sun.  Convection  currents  might 
be  violent  enough  and  might  reach  deep  enough  to  bring  to 
the  surface  the  requisite  amount  of  heat.  In  order  to  get 
a  quantitative  idea  of  the  requirements  suppose  that  essen- 
tially all  of  the  sun's  radiation  is  from  a  layer  of  the  photo- 
sphere, of  average  density  one  tenth,  500  miles  thick.  Sup- 
pose its  specific  heat  is  unity.  At  the  rate  at  which  the  sun 
radiates,  the  temperature  of  this  layer  would  decrease  one 
degree  Fahrenheit  in  1.6  hours  if  fresh  energy  were  not  sup- 
plied from  below.  Hence  the  requirements  do  not  seem  to 
be  unreasonably  severe. 

In  a  body  as  nearly  opaque  as  the  sun  seems  to  be,  radiation 
probably  is  of  no  importance  in  the  escape  of  heat  from  the 
deep  interior  to  the  surface  layers. 

229.  Sun  Spots.  —  The  most  conspicuous  markings  ever 
seen  on  the  sun  are  relatively  dark  spots  which  occasionally 
appear  in  the  photosphere  and  last  from  a  few  days  up  to 
several  months,  with  an  average  duration  of  a  month  or  two. 
The  typical  spot  consists  of  a  round,  relatively  black  nucleus, 
called  the  umbra,  and  a  surrounding  less  dark  belt  called  the 
penumbra,  Fig.  144.  The  penumbra  is  made  up  of  con- 
verging filaments,  or  "  willow  leaves,"  of  brighter  material, 
which  look  as  though  the  intensely  luminous  photospheric 


382      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  229 

columns  were  tipped  over  so  as  to  make  their  sides  visible. 
The  umbra  and  penumbra  do  not  gradually  merge  into  each 
other,  and  likewise  the  penumbra  and  surrounding  photo- 
sphere have  a  fairly  definite  line  of  separation. 

The  umbra  of  a  sun  spot  may  be  anywhere  from  500  to 
50,000  miles  across ;  the  diameter  of  the  penumbra  may  be 
as  great  as  200,000  miles.  When  the  spots  are  of  these 
dimensions  they  can  be  seen  simply  with  the  aid  of  a  smoked 
glass  to  reduce  the  glare  of  the  sun.     The  Chinese  claim  to 


Fig.   144.  —  Great  sun  spot  of  July  17,  1905.     Photographed  by  Fox  with  the 
40-inch  telescope  of  the  Yerkes  Observatory. 

have  records  of  observations  of  sun  spots  made  centuries 
before  their  discovery  by  Galileo  in  1610. 

The  umbra  of  a  sun  spot  is  dark  only  in  comparison  with 
the  glowing  photosphere  which  surroimds  it,  for  a  calcium 
hght  projected  on  it  appears  black.  In  fact,  it  sometimes 
shows  many  details  of  darker  spots  and  brighter  streaks  which 
most  often  appear  shortly  before  it  breaks  up.  In  the 
neighborhood  of  spots  the  brightness  of  the  photosphere  is 
usually  above  the  average,  and  there  are  nearly  always  in 
their  vicinity  very  bright  elevated  masses  of  calcium  which 
constitute  the  faculce.     These  faculae   are    especially  con- 


CH.  XI,  230]  THE   SUN  383 

spicuous  when  near  the  sun's  apparent  margin,  or  limb,  as 
it  is  called,  for  in  these  regions  the  photosphere  is  greatly 
dimmed  by  the  extensive  absorbing  material  through  which 
its  rays  must  pass,  while  on  the  other  hand  the  faculse  pro- 
ject out  through  the  absorbing  material  and  shine  with  but 
shghtly  diminished  luster. 

230.  The  Distribution  and  Periodicity  of  Sim  Spots.  — 
Sun  spots  are  rarely  seen  except  in  two,  belts  extending  from 
latitude  6°  to  latitude  35°  on  each  side  of  the  sun's  equator. 
Moreover,  they  are  not  always  equally  numerous.  For 
three  or  four  years  they  appear  with  great  frequency,  then 
they  become  less  numerous  and  dechne  to  a  minimum  for 
three  or  four  years,  after  which  they  are  more  numerous 
again.  The  interval  from  maximum  number  to  maximum 
number  averages  about  11.11  years,  though  the  period  varies 
from  about  7  years  to  more  than  16  years.  When  a  period 
is  short  the  maximum  which  follows  it  is  very  marked,  as 
though  a  large  amount  of  sun-spot  activity  had  been  crowded 
into  a  short  interval ;  on  the  other  hand,  when  a  maximum 
is  delayed  it  is  below  normal  in  activity. 

There  is  a  connection  between  the  positions  of  sun  spots 
and  their  numbers,  first  pointed  out  by  Schwabe  in  1852. 
After  a  sun-spot  maximum  has  passed,  the  spots  appear 
year  after  year  for  about  five  years,  on  the  average,  in  suc- 
cessively lower  latitudes,  and  they  are  continually  less 
numerous.  At  the  sixth  year  a  few  are  still  visible  in  about 
latitudes  ±  6°,  and  a  new  cycle  starts  in  latitudes  about  ±  35°. 
After  this  the  spots  in  the  low  latitudes  disappear,  those  in 
the  higher  latitudes  increase  in  numbers,  but  gradually  de- 
scend in  latitude  until  the  mg,ximum  activity  is  reached  in 
latitudes  ±  16°.  The  areas  covered  by  spots  in  years  of 
maximiun  activity  are  from  15  to  45  times  those  covered  in 
years  of  minimum  activity.  The  results  from  1876  to  1902 
are  shown  in  Fig.  145. 

Since  accurate  records  of  the  numbers  and  dimensions  of 
sun  spots  have  been  kept,  the  sun's  southern  hemisphere 


384      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  230 

has  been  somewhat  more  active  than  the  northern.  For  the 
period  from  1874  to  1902,  57  per  cent  of  the  total  spot  area 
was  in  the  southern  hemisphere  of  the  sun  and  only  43  per 
cent  in  the  northern.  That  is,  the  activity  in  the  southern 
hemisphere  was  about  one  third  greater  than  that  in  the 


j55)l8;7|ie78|ie?9llSM|i8Bc|lMl|lB83|ie84|i«a5|lBe6|ie«l|PMS|lSBs|l650|l«Sl|l»32|      '  |l8S4|l855|l8S6|lBs7]lBSe|l833|l300|lS0l  |l30:|.^ 


}lB77[lB7a|lfl73|lBaoll6Bl[lBB2[lBB3[l8M[l885|l8Ba[l887[l88B}lBS9|ia90[        ||ia52|lBB3[lBB4|lBB5|lB96[ie37[l6B8[l839|l90o]l90l 


Fig.  145.  —  Distribution  and  magnitudes  of  sun  spots  for  the  period  from 
1876  to  1902  (Maunder). 


northern.    Whether  this  difference  is  permanent  and  what  it 
means  cannot  at  present  be  determined. 

231.  The  Motions  of  Sun  Spots.  — Individual  sun  spots 
may  drift  both  in  latitude  and  in  longitude,  and  they  often 
have  complicated  and  violent  internal  motions.  As  a  rule, 
those  spots  whose  latitudes  are  less  than  20°  drift  slowly 
toward  the  sun's  equator,  and  those  which  are  in  higher 
latitudes  drift  away  from  it.  When  two  spots  are  near  to- 
gether they  are  generally  on  the  same  parallel  of  latitude. 
The  spot  which  is  ahead  usually  moves  forward  with  respect 
to  the  sun's  surface,  while  the  one  which  is  behind  lags  con- 


CH.  XI,  231]  THE    SUN  385 

tinually  farther  in  the  rear.  If  a  large  spot  divides  into  two 
components,  they  generally  recede  from  each  other,  some- 
times at  the  rate  of  1000  miles  an  hour. 

Sun  spots  sometimes  have  spiral  motions,  but  until 
recently  the  phenomenon  was  thought  to  be  hardly  char- 
acteristic because  it  was  observed  in  only  a  small  percentage 
of  cases.  Hale's  invention  of  the  spectroheliograph  (Art. 
237)  furnished  a  new  and  powerful  means  of  stud3dng  solar 
phenomena,  and  it  has  led  in  recent  years  to  a  discovery 
of  great  interest  and  importance  in  this  connection. 

In  1908  Hale  proved  the  existence  of  magnetic  fields  in 
the  high  levels  of  sun  spots.  One  may  well  wonder  how  such 
a  result  could  be  established,  since  we  receive  only  light  and 
heat  from  the  sun.  Naturally  it  must  be  done  from  the 
characteristics  of  the  radiant  energy  which  the  sun  sends  to 
the  earth.  About  1896  Zeeman  found  that  most  spectral 
lines  are  doubled,  or  at  least  widened,  when  observed  along 
the  lines  of  force  of  a  magnet,  and  that  the  two  components 
are  circularly  polarized  in  opposite  directions.  Hale  ex- 
amined the  widened  spectral  lines  belonging  to  sun  spots 
^nd  proved  that  they  have  the  properties  of  spectral  lines  in 
a  magnetic  field.  Then  he  took  up  the  question  of  the 
origin  of  the  magnetic  fields.  It  was  shoAvn  by  Rowland  in 
1876  that  static  electric  charges  in  revolution  produce  elec- 
tromagnetic effects  Uke  those  produced  by  electric  currents. 
Consequently  Hale  concluded  that  the  magnetic  fields  in 
sun  spots  are  due  to  vortical  motions  of  particles  carrying 
static  electric  charges,  and  the  explanation  is  almost  cer- 
tainly correct. 

More  recently  the  whole  sun  has  been  found  to  be  involved 
in  a  magnetic  field  whose  poles  agree  approximately  with 
its  poles  of  rotation;  it  may  be  analogous  to  that  which 
envelopes  the  earth.  Schuster  has  suggested  that  the  mag- 
netic states  of  the  earth  and  sun  may  be  a  consequence  of 
their  rotations,  and  that  all  rotating  bodies  must  be  magnets. 

Hale's  discovery  is  a  proof  of  cyclonic  motion  in  the 
2c 


386      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  231 

upper  parts  of  sun  spots.  Unlike  cyclones  on  the  earth,  the 
direction  of  motion  in  a  hemisphere  is  not  always  the  same. 
Hale  found  numerous  examples  where  two  spots  seemed  to 
be  connected,  one  having  one  polarity  and  the  other  the 
opposite  (Fig.  146).  It  has  been  suggested  they  are  the 
two  ends  of  a  cyhndrical  whirl.  This  idea  is  confirmed,  at 
least  to  some  extent,  by  the  fact  that,  so  far  as  observational 
evidence  goes  at  present,  when  two  spots  are  near  together, 
they  always  have  opposite  polarity.     Another  remarkable 


Fig.  146.  —  Sun  spots  having  opposite  polarity.     Photographed  cd  the  Mi. 
Wilson  Solar  Observatory  with  the  spectroheliograph  (Hale) . 

fact  is  that  if  two  neighboring  spots  are  in  the  northern  hemi- 
sphere of  the  sun,  the  one  which  is  ahead  has  a  counter- 
clockwise vortical  motion,  while  the  motion  in  the  other  is 
in  the  opposite  direction.  The  conditions  are  the  opposite 
in  the  sun's  southern  hemisphere. 

Evershed,  in  India,  announced  in  1909  that  at  the  lowest 
visible  levels  there  is  radial  motion  outward  from  spots 
parallel  to  the  surface  of  the  sun.  More  recently  St.  John, 
at  the  Mt.  Wilson  Solar  Observatory  (Fig.  147),  has  made 
extensive  studies  of  the  motions  in  sun  spots  with  the  ad- 


CH.  XI,  231] 


THE    SUN 


387 


vantage  of  most  powerful  instruments,  and  he  concludes 
that  at  the  lower  levels  there  is  motion  radially  outward 
from  spot  centers,  at  levels  about  2500  miles  higher  there  is 
no  horizontal  motion,  and  in  the  high  levels  of  the  chromo- 
sphere (10,000  to  15,000  miles)  the  motion  is  inward  toward 
the  centers  of  the  spots.  This  suggests  that  spots  are  pro- 
duced-by  cooler  gases  from  high  levels  rushing  in  toward  a 
center,  descending  some  thousands  of  miles,  and  then  spread- 
ing out  at  lower  levels,  but  the  consideration  of  the  quality 
and  quantity  of  the  materials  involved  in  the  two  move- 


FlG.  147.  —  The  Mt.  Wilson  Solar  Observatory  of  the  Carnegie  Institution 
of  Washington.     Pasadena,  California. 

ments,  together  with  their  kinetic  energies,  led  St.  John  to 
the  conclusion  that  the  material  flowing  inward  and  down- 
ward by  no  means  equals  that  flowing  outward  at  lower 
levels  from  the  axes  of  spots.  He  believes,  rather,  that  a 
spot  is  formed  by  currents  ascending  from  the  sun's  interior 
and  spreading  out  just  above  the  photosphere.  The  in- 
rushing  and  descending  chromospheric  material  is  a  second- 
ary result  of  the  primary  currents.  The  spots  are  dark  be- 
cause the  expanding  gases  of  which  they  are  composed  are 
cooler  than  those  which  constitute  the  photosphere. 

Independent  evidence  of  a  conclusive  character  shows 
that  spots  are  cooler  than  the  ordinary  photosphere.     There 


388     ■  AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  ^6l 

is  evidence  from  the  so-called  enhanced  spectral  Unes  which 
has  been  brought  out  by  Hale,  Adams,  and  Gale ;  the  lines 
in  the  spectrmn  of  spots  are  related  to  those  in  the  spectnmi 
of  the  remainder  of  the  sun  just  as  the  spectra  with  low  tem- 
peratures in  the  electric  furnace  are  related  to  those  with 
high  temperatures;  and  finally,  the  spectra  of  spots  con- 
tain flu  tings,  or  bands,  which  are  believed  to  be  due  to  ab- 
sorption by  chemical  compounds  which  would  be  broken  up 
into  their  constituent  elements  in  the  higher  temperatures  of 
the  photosphere.  • 

232.  The  Rotation  of  the  Sun.  —  The  rotation  of  at  least 
that  part  of  the  sun  in  which  the  spots  occur  can  be  found 
from  their  apparent  transits  across  its  disk.  The  first 
systematic  investigation  of  the  sun's  rotation  was  made  by 
Carrington  and  Spoerer  about  the  middle  of  the  nineteenth 
century.  They  found  that  the  sun  rotates  from  west  to  east 
about  an  axis  inclined  7°  to  the  perpendicular  to  the  plane 
of  the  echptic.  The  sun's  axis  points  to  a  position  whose 
right  ascension  and  declination  are  respectively  18  h.  44  m. 
and  -|-46°,  which  is  almost  exactly  midway  between  Vega 
and  Polaris.  The  period  of  the  solar  rotation  depends  upon 
the  latitude.  Spots  near  the  sun's  equator  complete  a  revo- 
lution in  about  25  days ;  in  latitude  30°,  in  about  26.5  days ; 
in  latitude  45°,  in  about  27.5  days ;  in  higher  latitudes  spots 
are  not  seen. 

Reference  has  already  been  made  to  the  facula;,  or  bright 
clouds,  which  are  especially  abundant  in  the  neighborhood 
of  sun  spots.  The  positions  of  the  faculaj  are  easily  de- 
termined on  photographs  of  the  sun,  and  from  photographs 
made  at  sufficiently  short  intervals  the  rotatioii  of  the  sun 
can  be  found.  This  method  has  given  results  in  accord  with 
those  obtained  from  observations  of  spots. 

The  remarkable  developments  of  spectroscopic  methods 
which  followed  Hale's  invention  of  the  spectroheUograph 
have  furnished  a  third  method  of  measuring  the  rotation  of 
the  sun.     By  its  use  bright  clouds  of  calcium  vapor,  called 


CH.  XI,  232] 


THE   SUN 


389 


flocculi  by  Hale,  and  both  bright  and  dark  floccuU  of  hydro- 
gen have  been  photographed.  The  rotation  of  the  sun  has 
been  determined  by  Hale  and  Fox  from  photographs  of 
floccuh. 

Finally,  the  rotation  of  the  sun  has  been  determined  by 
the  Doppler-Fizeau  effect.  One  hmb  of  the  sun  at  the  equa- 
tor approaches  the  earth  at  the  rate  of  1.3  miles  per  second, 
while  the  other  recedes  at  the  same  velocity.  The  spectro- 
scopic method  is  so  highly  developed  that  it  not  only  gives 
the  rate  of  rotation  of  the  sun  approximately,  but  it  shows 
that  the  period  is  shorter  at  the  equator  than  it  is  in  higher 
latitudes. 

The  results  for  the  periods  of  rotation  of  the  sun  by  the 
various  methods  are  given  in  the  following  table,  in  which 
the  results  are  expressed  in  mean  solar  days : 

Table  X 


Latitude 

Sun  Spots 

FaCUIi^B 

Calcium 

FliOCCULI 

Hydrogen 
Flocculi 

Doppler- 
Fizeau 
Method 

0°  to    5° 
5   to  10 
10  to  15 
15  to  20 
20   to  25 
25   to  30 
30  to  35 

25.00 
25.09 
25.26 
25.48 
25.75 
26.09 
26.47 

24.73 
24.79 
25.12 
25.33 
25.37 
25.64 
26.47 

24.76 
24.98 
25.17 
25.48 
25.73 
25.77 
26.18 

25.7 
25.0 
24.7 
24.8 
24.5 
24.5 
24.2 

24.67 
24.86 
25.12 
25.44 
25.81 
26.20 
26.67 

By  the  Doppler-Fizeau  method  Adams  found  the  periods 
of  rotation  of  the  sun  in  latitudes  45°,  60°,  and  74°,  to  be 
respectively  28.1,  31.3,  and  32.2  days. 

The  reason  that  the  sun  rotates  in  its  peculiar  manner  is 
not  certainly  known,  though  EUiott  Smith  has  attempted 
to  show  that  the  more  rapid  rotation  of  the  equatorial  zone 
is  an  inevitable  consequence  of  the  contraction  of  a  rotating 
mass  of  gas.  The  question  deserves  further  quantitative 
examination. 


390      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  232 

Under  the  hypothesis  that  the  sun  is  a  mixture  of  fluids 
in  equilibrium,  Wilsing,  Sampson,  and  Wilczynski  have 
reached  the  conclusion  from  hydrodynamical  considerations 
that  cylindrical  layers  of  it  rotate  with  the  same  speed. 
According  to  this  view  the  outermost  cyhnder,  which  in- 
cludes only  the  equatorial  zone,  rotates  fastest,  and  suc- 
cessive cyhnders  toward  the  axis  rotate  more  and  more  slowly. 
It  is  supposed  that  this  condition  is  inherited  from  some 
primitive  state  and  that  friction  has  not  yet  reduced  the 
rotation  to  uniformity.  Wilczynski  showed  that  friction 
between  the  different  layers  would  not  wear  down  the  dif- 
ferences of  motion  appreciably  in  many  milhons  of  years. 
But  he  neglected  the  convection  currents  which  must  cer- 
tainly exist  to  great  depths  and  which  would  quickly  destroy 
the  supposed  different  rotations  in  different  cyhnders. 
Notwithstanding  these  difficulties,  no  other  theory  at  present 
is  more  satisfactory  than  that  the  sun's  pecuUar  rotation  has 
been  inherited  from  more  extreme  conditions  which  pre- 
vailed in  the  remote  past. 

233.  The  Reversing  Layer.  —  Newton  began  the  analysis 
of  fight  by  passing  it  through  a  small  circular  opening.  In 
1802  Wollaston  passed  the  fight  from  the  sun  through  a 
narrow  sfit,  instead  of  a  pinhole,  and  found  that  the  solar 
spectrum  was  crossed  by  7  dark  fines.  In  a  few  years  the 
subject  was  taken  up  by  Fraunhofer,  who  soon  foimd  that 
the  spectrum  was  crossed  by  an  immense  number  of  dark 
fines.  In  1815  he  mapped  324  of  them,  and  they  have  since 
been  known  as  "  Fraunhofer  lines."  A  greatly  improved 
map  of  these  fines  was  made  by  Kirchhoff  in  1862,  and  still 
another  by  Angstrom  in  1868.  In  1886  Langley  mapped 
the  solar  spectrum  with  the  aid  of  his  bolometer  far  into  the 
infra-red  region,  and  in  1886,  1889,  and  1893  Rowland  pub- 
fished  extensive  and  very  accurate  maps  from  measurements 
of  the  positions  and  characteristics  obtained  with  his  power- 
ful grating  spectroscope.  In  1895  Rowland  pubfished  his 
great  "  PreUminary  Table  of  Solar  Spectrum  Wave  Lengths," 


CH.  XI,  233]  ■ 


THE   SUN 


391 


containing  the  results  for  about  14,000  spectral  lines.  A 
portion  of  the  solar  spectrum  is  shown  in  Fig.  148  with  a 
bright-Une  comparison  spectrum  above. 

The  spectrum  of  the  sun  is  continuous  except  for  the  very 
nimierous  dark  hues  which  cross  it.  Therefore,  in  accord- 
ance with  the  third  law  of  spectrum  analysis,  there  is  be- 
tween the  photosphere  and  the  observer  cooler  gas,  and  its 
constitution  can  be  determined  from  the  relations  among  the 
dark  hues  and  from  their  positions.  The  Unes  prove  the 
existence  of  sodiimi,  iron,  and  other  heavy  metals  in  this 
intervening  gas,  and  since  they  cannot  remain  in  the  gaseous 
state  in  our  own  atmosphere  they  must  be  in  that  of  the  sun. 


Fig.  148.- 


-  Portion  of  solar  spectrum  below  with  a  Titanium  comparison 
spectrum  above. 


This   absorbing  material   which   overUes   the  photosphere 
constitutes  the  reversing  layer. 

If  the  reversing  layer  could  be  viewed  not  projected  against 
the  brilliant  photosphere,  it  would  give  a  spectrum  of  bright 
lines  exactly  at  the  places  occupied  by  the  dark  lines  under 
the  conditions  as  they  normally  exist.  At  the  total  eclipse 
of  the  sun  in  1870,  Young  placed  the  slit  of  his  spectroscope 
tangent  to  the  limb  of  the  sun.  Just  as  the  moon  cut  off  the 
last  of  the  photosphere  the  spectrum  suddenly  flashed  out 
in  bright  Unes  where  an  instant  before  the  dark  ones  had 
been.  Since  1895,  dining  nearly  every  total  eclipse  of  the 
sun,  this  "  flash  spectrum  "  has  been  photographed,  and 
there  is  no  doubt  that  the  positions  of  its  lines  are  identical 
with  those  of  the  corresponding  dark  Fraunhof  er  lines.  From 
the  duration  of  their  appearance  as  bright  Unes  and  the  known 
rate  at  which  the  moon  apparently  passes  across  the  disk  of 


392      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  233 

the  sun,  it  has  been  found  that  the  reversing  layer  is  500  or 
600  nailes  deep. 

As  a  rule  the  effect  of  pressure  on  an  absorbing  gas  is  to 
cause  the  dark  hues  to  shift  slightly  toward  the  red  end  of 
the  spectrum.  Extensive  studies  by  various  astronomers  of 
the  displacements  of  the  Fraunhofer  lines  have  led  to  the 
conclusion  that  the  pressure  of  the  reversing  layer,  even  at 
its  lower  levels,  does  not  exceed  5  or  6  times  that  of  the 
earth's  atmosphere  at  sea  level.  This  is  a  very  remarkable 
result  in  view  of  the  great  extent  of  the  sun's  atmosphere 
and  the  fact  that  gravity  at  the  surface  of  the  sun  is  nearly 
28  times  as  great  as  it  is  at  the  surface  of  the  earth.  Pos- 
sibly electrical  repulsion  from  the  sun  and  light  pressure 
partly  offset  the  great  surface  gravity  of  the  sun. 

234.  Chemical  Constitution  of  the  Reversing  Layer.  — 
Of  the  14,000  hues  in  Rowland's  spectrum  about  one  third 
are  due  to  the  absorption  by  the  earth's  atmosphere,  and 
the  remainder  are  produced  by  the  sun's  reversing  layer 
and  chromosphere.  By  comparing  the  positions  of  the  lines 
of  the  sun's  spectrum  with  those  given  by  the  various  ele- 
ments in  laboratory  experiments,  it  is  possible  to  infer  the 
chemical  constitution  of  the  material  which  produces  the 
absorption.  In  this  manner  38  elements  are  known  cer- 
tainly to  exist  in  the  sun,  but  more  than  6000  of  the  lines 
mapped  by  Rowland  have  not  as  yet  been  identified  as 
belonging  to  any  element. 

The  presence  of  iron  is  estabhshed  by  more  than  2000 
line  coincidences,  carbon  by  more  than  200,  calcium  by  more 
than  75,  magnesium  by  20,  sodium  by  11,  copper  by  2,  and 
lead  by  1.  It  will  be  noticed  that  nearly  all  the  elements 
in  the  table  which  follows  are  metals,  the  exceptions  being 
hydrogen,  helium,  carbon,  and  oxygen.  Qn  the  other  hand, 
a  number  of  heavy  metals,  such  as  gold  and  mercury,  are 
missing.  The  following  table  gives  the  elements  found  in  the 
sun  and  their  atomic  weights : 


CH.  XI,  234] 


THE   SUN 


393 


Table  XI 


Element 

Atomic  Weight 

Element 

Atomic  Weight 

Hydrogen     . 

1 

Copper  .^   . 

64 

Helium    . 

4 

Zinc  .     .     . 

65 

Gluoinum 

9 

Germanium 

72 

Carbon    .     . 

12 

Strontium  .     . 

88 

Oxygen    . 

16 

Yttrium 

■     89 

Sodium   . 

23 

Zirconium  .     . 

91 

Magnesium . 

24 

Niobium      .     . 

93 

Aluminum"  .     . 

27 

Molybdenum 

96 

Silicon     .     .     . 

28 

Rhodium     . 

103 

Potassium 

39 

Palladium  .     . 

107 

Calcium  .     . 

40 

Silver 

108 

Scandium 

44 

Cadmium   .     . 

112 

Titanium     .     . 

48 

Tin    .     .     . 

119 

Venadium    .     . 

51 

Barium  .     .     . 

137 

Chromium   . 

52 

Lanthanum 

139 

Manganese  . 

55 

Cerium  .     .     . 

140 

Iron    .     .     . 

56 

Neodymium    . 

144 

Nickel 

59 

Erbium  . 

168 

Cobalt 

59 

Lead .     .     . 

207 

While  the  presence  of  the  spectral  lines  of  an  element  proves 
its  existence,  their  absence  does  not  show  that  it  is  not  pres- 
ent. ,  In  the  first  place,  heavy  elements,  like  gold,  mercury, 
and  platinum,  would  probably  sink  far  below  the  level  of 
the  reversing  layer;  and  consequently  would  give  no  hues  in 
the  solar  spectrum.  Then,  again,  the  characteristic  spectra 
of  some  of  the  elements,  particularly  non-metals,  are  sup- 
pressed by  the  presence  of  some  other  elements,  particularly- 
metals.  Sometimes  the  spectrum  of  an  element  is  entirely 
obliterated  by  the  presence  of  a  small  percentage  of  another 
element.  This  may  be  the  explanation  of  the  fact  that  the 
spectra  of  fluorine,  chlorine,  bromine,  iodine,  sulphur,  sele- 
nium, tellurium,  nitrogen,  phosphorus,  arsenic,  antimony, 
and  boron  are  not  found  in  the  sun,  although  most  of  these 
elements  occur  abundantly  in  the  earth.  Some  elements 
have  spectra  that  change  radically  with  alterations  in  their 


394      AN   INTRODUCTION   TO  ASTRONOMY    [oh.  xi,  234 

conditions  of  temperature,  pressure,  and  electrical  excitation. 
One  of  these  elements  is  oxygen,  which  was  long  sought  for 
in  the  sun  before  it  was  certainly  found.  Of  course,  the  proof 
of  its  existence  was  complicated  by  the  fact  that  it  occurs 
in  abundance  in  the  earth's  atmosphere.  Finally,  as  Lock- 
yer  suggested,  some  of  the  so-called  elements  may  be  in 
reahty  compounds  which  are  broken  up  under  the  extreme 
conditions  of  temperature  prevailing  in  the  sun,  and  their 
characteristic  spectra  may  be  in  this  manner  destroyed. 

The  reversing  layer  is  undoubtedly  constantly  receiving 
material  from  below  and  above,  and  therefore  it  is  safe  to 
conclude  that  its  composition  is  not  qualitatively  different 
from  that  of  the  remainder  of  the  sun.  It  is  interesting 
that  nearly  40  terrestrial  elements  are  found,  for  it  points 
strongly  to  the  conclusion  that  the  sun  and  the  earth  have 
had  a  common  origin. 

The  distribution  of  the  elements  in  distance  above  the 
sun's  photosphere  was  determined  by  Mitchell  from  excel- 
lent photographs  of  the  flash  spectrum  which  he  secured  in 
the  eclipse  of  1905,  and  by  St.  John  from  considerations  of 
the  Doppler-Fizeau  effect.  On  the  whole  the  lighter  ele- 
ments extend  to  high  altitudes  while  the  heavier  elements 
are  confined  to  the  lower  levels.  A  peculiar  exception  is 
that  calcium,  whose  atomic  weight  is  40,  extends  in  abun- 
dance up  into  the  chromosphere  10,000  miles,  even  as  high 
as  hydrogen.  Iron  and  the  heavier  metals  are  found  only 
down  in  the  reversing  layer. 

235.  The  Chromosphere.  —  As  has  been  stated,  the  chro- 
mosphere is  a  gaseous  envelope  around  the  sun  above  the 
reversing  layer  whose  depth  is  from  5000  to  10,000  miles.  It 
gets  its  scarlet  color  from  the  incandescent  hydrogen  and 
calcium  of  which  it  is  largely  composed. 

The  spectrum  of  the  chromosphere  consists  of  many  lines, 
some  of  which  are  permanent  while  others  come  and  go. 
The  permanent  hnes  are  due  mostly  to  hydrogen,  helium,  and 
calcium;  the  intermittent  lines  are  due  to  many  elements. 


CH.  XI,  236]  THE   SUN  395 

which  seem  to  have  been  temporarily  thrown  up  into  it 
through  the  reversing  layer. 

The  existence  of  the  element  helium  was  first  inferred  from 
the  presence  of  a  bright  yellow  line  in  the  solar  spectrum  near 
the  two  yellow  lines  of  sodium.  It  is  universally  prevalent 
in  the  chromosphere,  giving  a  bright  line  when  the  sun  is 
eclipsed,  or  at  any  time  when  the  sUt  of  the  spectroscope  is 
put  on  the  chromosphere  parallel  to  the  sun's  limb.  For 
some  unknown  reason  helium  does  not  give  a  dark-line  ab- 
sorption spectrum  when  the  light  from  the  photosphere 
passes  through  it.  This  seems  to  be  a  direct  contradiction 
to  the  third  law  of  spectrum  analysis,  which  holds  true  in 
all  other  known  cases.  But  helium  is  a  very  remarkable 
element  in  several  other  respects.  Next  to  hydrogen,  it 
has  the  lowest  atomic  weight,  it  is  very  inactive,  and  enters 
into  no  known  chemical  combinations  with  other  elements, 
it  has  the  lowest  known  refractive  index,  it  is  an  excellent 
conductor  of  electricity,  its  rate  of  diffusion  is  15  times  its 
theoretical  value,  its  solubility  in  water  is  nearly  zero,  and  it 
is  liquefied  only  with  the  utmost  difficulty.  It  has  already 
been  explained  that  helium  is  one  of  the  products  of  the  dis- 
integration of  uranium,  radium,  and  other  radioactive  sub- 
stances. It  was  not  discovered  on  the  earth  until  1895,  when 
Ramsay,  on  examining  the  spectrum  of  the  mineral  clevite, 
found  the  yellow  spectral  line  of  helium. 

236.  Prominences.  —  Vast  eruptions,  called  prominences, 
shoot  up  from  the  sun's  photosphere  through  its  chromo- 
sphere to  heights  ranging  from  20,000  miles  up  to  300,000 
miles,  or  even  to  greater  elevations  in  extreme  cases.  One 
80,000  miles  in  height  is  shown  in  Fig.  149.  They  usually 
occur  in  the  neighborhood  of  sun  spots  and  are  never  seen 
near  the  sun's  poles.  They  leap  up  in  jets  and  flames,  often 
changing  their  appearance  greatly  in  the  course  of  10  or  15 
minutes,  as  is  shown  in  Fig.  150.  Their  velocity  of  ascent 
is  frequently  100  miles  per  second  and  sometimes  exceeds 
200  miles  per  second. 


396      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  236 

If  eruptive  prominences  should  leave  the  photosphere  with 
a  velocity  of  more  than  380  miles  per  second,  and  if  they 
should  suffer  no  resistance  from  the  reversing  layer  and 
chromosphere,  they  would  escape  entirely  from  the  sun  and 
pass  out  beyond  the  planets  to  the  distances  of  the  stars. 
It  is  very  difficult  to  account  for  their  great  velocities.  No 
satisfactory  theory  has  been  developed  for  explaining  how 
such  violent  explosive  forces  are  long  held  in  restraint  and 


Fig.  149.  —  Solar  prominence,  August  21,   1909,  reaching  to  a,  height  of 
80,000  miles.     Photographed  at  the  Mt.  Wilson  Solar  Observatory. 

then  suddenly  released.  Perhaps  under  the  extreme  condi- 
tions of  temperature  and  pressure  prevailing  in  the  interior 
of  the  sun,  all  elements,  like  radium  under  terrestrial  con- 
ditions, explode  because  of  their  subatomic  energies.  Julius 
has  maintained  that  the  prominences  may  be  mirage-like 
appearapces  due  to  unusual  refraction,  and  that  they  are  not 
actual  eruptions  from  the  sun  as  they  seem  to  be.  But  their 
velocities  are  determined  both  from  their  motion  perpen- 
dicular to  the  line  of  sight  when  they  are  seen  on  the  sun's 
limb,  and  also  from  spectral  line  displacements  in  accord- 


CH.  XI,  236]  THE   SUN  397 

ance  with  the  Doppler-Fizeau  principle,  and  it  seems  very- 
improbable  that  they  are  not  real. 

The  spectra  of  eruptive  prominences  show  many  lines, 
especially  in  the  lower  levels.  In  them  the  bright  hnes  of 
sodium,  magnesium,  iron,  and  titanium  are  conspicuous, 
while   those   of   calcium,    chromium,    and   manganese   are 


Fig.  150.  —  Changes  in  a  solar  prominence  in  an  interval  of  ten  minutes. 
Photographed  by  Slocum  at  the  Yerkes  Observatory. 

generally  found.  In  the  higher  levfels  calcium  is  the  pre- 
dominating element,  a  remarkable  fact  in  view  of  its  atomic 
weight  of  40. 

Prominences  were  formerly  observed  only  when  the  sun 
was  totally  eclipsed,  for  at  other  times  the  illumination  of 
the  sky  made  them  altogether  invisible.  But  since  the  de- 
velopment of  the  spectroscope  they  can  be  observed  at  any 
time.     If  the  light  from  the  limb  of  the  sun  is  passed  through 


398      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  236 

the  spectroscope,  the  continuous  illumination  of  the  earth's 
atmosphere  is  spread  out  and  correspondingly  enfeebled; 
on  the  other  hand,  the  light  from  the  prominences  consists 
of  single  colors  and  is  not  diminished  in  intensity  by  passing 
through  the  spectroscope.  Consequently,  if  the  dispersion 
is  sufficient,  the  atmospheric  illumination  is  reduced  until 
the  prominences  become  visible. 

Not  all  the  prominences  are  eruptive.  Besides  those 
which  burst  out  suddenly,  rising  to  great  heights  and  soon 
disappearing  or  subsiding  again,  there  are  others,  called 
quiescent  prominences,  which  spread  out,  like  the  tops  of 
banyan  trees,  with  here  and  there  a  stem  reaching  below. 
They  often  develop  far  above  the  surface  of  the  sun,  without 
apparent  connections  with  it,  and  seem  to  be  due  to  material 
which  for  some  mysterious  reason  suddenly  becomes  visible. 
They  rest  qui'etly  at  great  altitudes,  somewhat  like  terrestrial 
clouds,  often  for  many  days,  notwithstanding  the  sun's 
gravity.  They  are  made  up  of  hydrogen,  hehum,  and 
calcium. 

237.  The  Spectroheliograph.  —  The  photosphere  radi- 
ates a  continuous  spectrum,  while  above  it  is  the  reversing 
layer  which  produces  the  dark  absorption  lines.  Some  of  the 
lines,  as  the  -fC-line  due  to  calcium,  are  broad  because  of  the 
great  extent  of  the  absorbing  layer.  Now,  calcium  is  abun- 
dant in  the  prominences,  and,  moreover,  it  shines  with  an 
intensity  greater  than  that  of  the  reversing  layer.  The  re- 
sult is  that  the  reversing  layer  makes  a  broad,  dark  hne,  say 
the  K-]ine,  and  above  it  is  more  luminous  calcium  in  a  rarer 
state  which  produces  a  narrow  bright  line  in  the  midst  of 
the  dark  one.     The  hne  is  said  to  be  "  doubly  reversed." 

The  spectroheliograph  is  an  instrument  invented  and  per- 
fected by  Hale  in  1891  for  the  purpose  of  photographing 
the  sun  with  the  light  from  a  single  element.  The  ideas  on 
which  it  depends  were  almost  simultaneously  developed  and 
applied  by  Deslandres.  In  this  instrument,  or  rather  com- 
bination of  instruments,  the  sunUght  is  passed  through  a 


CH.  XI,  237] 


THE   SUN 


399 


spectroscope  and  is  spread  out  into  a  spectrum.  The  in- 
line, which  is  most  frequently  used,  is  doubly  reversed  in 
the  regions  of  faculse  and  prominences.  All  the  spectrum 
is  cut  off  by  an  opaque  screen  except  the  bright  part  of  the 
if-line  which  passes  through  a  second  narrow  slit.  That  is, 
the  only  light  which  passes  through  both  slits  is  the  calcium 
light  from  that  portion  of  the  sun's  image  which  falls  on  the 
first  sht  of  the  spectroscope.  In  Fig.  151,  S  is  the  image 
of  the  sun  at  the  focal  plane  of  the  telescope,  A  is  the  slit 
of  the  spectroscope  (the  prisms  are  not  shown),  T  is  the 
spectrum  which  falls  on  the  screen  B,  Risa  slit  in  the  screen 
B  which  is  adjusted  so  that  it  admits  the  bright- center  of 


Fig.   151. — ^The  spectroheliograph. 


the  doubly  reversed  if -line,  and  P  is  a  photographic  plate  on 
which  the  i^C-line  falls.  The  apparatus  is  so  made  that  the 
slit  A  may  be  moved  across  the  image  of  the  sun  S,  and  the 
sHt  R  simultaneously  moved  so  that  the  K-line  falls  on 
successively  different  parts  of  the  photographic  plate  P.  In 
this  manner  a  photograph  of  the  hot  calcium  vapors  which 
lie  above  the  reversing  layer  may  be  obtained ;  such  a  photo- 
graph is  shown  in  Fig.  152.  Some  other  spectral  lines  have 
also  been  used  in  this  way.  For  example,  two  photographs 
of  a  spot  with  the  so-called  i?-line  are  shown  in  Fig.  153. 

The  width  of  a  spectral  line  depends  upon  the  density  of 
the  gas  which  is  emitting  the  light.  Suppose  a  thick  layer 
of  calcium  gas  which  is  rare  at  the  top  and  denser  at  the  bot- 
tom gives  a  bright  K-line.     The  central  part  will  be  due  to 


400      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  237 

light  coming  from  all  depths,  particularly  from  the  higher 
layers  where  absorption  is  unimportant.  On  the  other 
hand,  the  marginal  parts  of  the  hne  will  be  due  to  light 


Fig.   152.  —  Spectroheliogram  of  the  sun  taken  with  the  doubly  reversed 
calcium  line.     Photographed  by  Hale  and  Ellerman  at  Yerkes  Observatory. 


coming  from  the  lower  levels  where  the  gas  is  denser.  Fol- 
lowing out  these  principles,  and  using  a  very  narrow  sHt, 
Hale  first  obtained  photographs  of  different .  levels  of  the 
solar  atmosphere. 


CH.  XI,  238]  THE   SUN  401 

238.  The  Corona.  —  During  total  eclipses  the  sun  is 
seen  to  be  surrounded  by  a  halo  of  pearly  light,  called  the 
corona,  extending,  out  200,000  or  300,000  miles,  while  some 
of  the  streamers  reach  out  at  least  5,000,000  miles.  So  far 
it  has  not  been  possible  to  find  any  observational  evidence 
of  the  corona  except  at  the  times  of  total  eclipses  of  the  sun. 
One  of  the  reasons  that  eclipses  are  of  great  scientific  in- 
terest is  that  they  afford  an  opportunity  of  studying  this 
remarkable  solar  appendage.  The  brief  duration  of  total 
eclipses  and  their  infrequency  have  made  progress  in  the 
researches  on  the   corona  rather  slow.     The  corona  is  not 


Fig.  153.  —  Spectroheliograms  of  a  sun  spot  with  the  doubly  reversed  H-line 
of  calcium.     Hale  and  EUerman,  Solar  Observatory,  Aug.  7  and  9,  1915. 

arranged  in  concentric  layers  like  an  atmosphere,  but  is 
made  up  of  comphcated  systems  of  streamers  (Fig.  154),  in 
general  stretching  out  radially  from  the  sun,  but  often  simply 
and  doubly  curved,  and  somewhat  resembling  aurorse. 
Many  observers  have  declared  that  its  finely  detailed  struc- 
ture resembles  the  Orion  nebula. 

The  coronal  streamers  often,  perhaps  generally,  have 
their  bases  in  the  regions  of  active  prominences,  but  excep- 
tions have  been  noted.  That  they  are  in  some  way  con- 
nected with  activity  on  the  sun  is  shown  by  the  fact  that-the 
form  qf  the  corona  changes  in  a  cycle  of  about  eleven  years, 
the  same  as  that  of  sun-spot  activity.  At  sun-spot  maxima, 
the  coronal  streamers  radiate  from  all  latitudes  nearly 
2d 


402      AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  238 


CH.  XI,  238]  ,  THE    SUN  403 

equally.  As  the  maxima  pass,  the  coronal  streamers  grad- 
ually withdraw  from  the  poles  of  the  sun  and  extend  out  to 
greater  distances  in  the  sun-spot  zones.  At  the  sun-spot 
minima,  the  corona  consists  of  short  rays  in  the  polar  regions, 
curved  away  from  the  solar  axis,  and  long  streamers  extend- 
ing out  in  the  equatorial  plane. 

The  spectroscope  shows  that  the  corona  emits  three  kinds 
of  light.  First,  there  is  a  small  quantity  which  is  known 
to  be  reflected  sunlight,  for  it  gives,  though  faintly,  the 
Fraunhofer  absorption  lines,  and  it  is  polarized.  Second, 
there  is  white  light  whose  source,  according  to  the  first  law 
of  spectrum  analysis,  must  be  incandescent  sohd  or  liquid 
particles.  Lastly,  there  is  a  bright-line  spectrum  whose 
source,  according  to  the  second  law  of  spectrum  analysis, 
must  be  an  incandescent  gas.  The  most  conspicuous  Une 
is  in  the  green  and  is  emitted  by  an  element,  called  coronium, 
which  is  not  yet  known  on  the  earth.  There  seems  to  be 
at  least  one  other  substance  present,  but  no  known  elements. 

According  to  present  ideas,  the  corona  consists  of  dust 
particles,  liquid  globules,  and  small  masses  of  gas  which 
are  widely  scattered.  From  the  amount  of  light  and  heat 
radiated,  and  from  the  temperature  which  masses  so  near  the 
sun  must  have,  Arrhenius  computed  that  there  is  one  dust 
particle,  on  the  average,  in  every  14  cubic  yards  of  the  corona. 
The  excessive  rarity  of  the  corona  is  shown  by  the  fact  that 
comets  have  plunged  through  hundreds  of  thousands  of  miles 
of  it  without  being  sensibly  retarded.  The  dust  particles 
and  Uquid  globules  give  the  reflected  hght ;  the  liquid,  the 
continuous  spectrum ;  and  the  gases,  the  bright-line  spectrum. 
The  form  of  the  corona  shows  that  its  condition  of  equilibrium 
is  not  at  all  similar  to  that  of  an  atmosphere  like  the  one  sur- 
rounding the  earth.  Its  increase  of  density  toward  the  sun 
is  inexplicably  slow,  though  doubtless  hght  pressure  and 
electrical  forces  are  opposed  to  gravity.  Its  radial  structure 
and  periodical  variation  in  general  form  are  without  satis- 
fg,ctory  explanation. 


404      AN    INTRODUCTION   TO   ASTRONOMY    [ch.  xi,  239 

239.  The  Eleven- Year  Cycle.  —  It  has  been  explained  that 
sun  spots  vary  in  frequency  and  distribution  on  the  sun's 
surface  in  a  period  averaging  a  little  more  than  11  years. 
There  are  a  number  of  other  phenomena  which  undergo 
changes  in  the  same  period. 

The  faculse  are  most  numerous  in  the  sun-spot  zones, 
although  they  occur  all  over  the  sun.  Both  their  number 
and  the  positions  of  the  zones  where  they  are  most  numerous 
vary  periodically  with  the  sun-spot  period.  This  is  quite 
to  be  expected,  for  the  sun  spots  and  the  faculae  are  both 
photospheric  phenomena. 

The  eruptive  prominences  are  frequent  in  the  sun-spot 
belts,  and  vary  in  position  with  them.  The  evidence  so  far 
also  shows  periodic  variations  in  their  numbers.  The  quies- 
cent prominences,  on  the  other  hand,  cluster  in  the  polar 
regions. 

The  coronal  types  clearly  vary  in  the  eleven-year  cycle, 
as  was  explained  in  the  preceding  article.  Doubtless  the 
total  solar  radiation  varies  to  some  extent  in  the  same  period, 
though  this  has  not  been  verified  observationally,  but  the 
time  is  now  ripe  for  the  investigation. 

The  spectra  of  sun  spots  vary  with  the  period  of  the  spots, 
but  the  Fraunhofer  lines  are  singularly  invariable. 

The  great  vibrations  which  so  powerfully  agitate  the 
sun  extend  to  the  earth  and  probably  to  the  whole  solar 
system.  It  has  long  been  known  that  both  the  horizontal 
and  vertical  components  of  the  earth's  magnetism  vary  in 
the  sun-spot  period,  and  that  magnetic  disturbances 
("  storms  ")  are  most  frequent  at  the  times  when  sun  spots 
are  most  numerous.  Likewise,  aurorse  occur  most  frequently 
at  the  epochs  of  great  sun-spot  activity.  In  fact,  magnetic 
storms  and  aurorse  never  occur  except  when  there  is  great 
activity  in  the  sun  in  the  form  of  sun  spots  or  prominences ; 
but  there  are  frequent  disturbances  on  the  sun  without 
accompanying  terrestrial  phenomena.  The  correlation  of 
these  phenomena  is  shown  in  Fig.  155. 


CH.  XI,  239] 


TH'E    SUN 


405 


The  first  suggested  explanation  of  magnetic  storms  on  the 
earth  was  that  the  sun  induces  changes  in  the  earth's  mag- 
netic state  by  sending  out  electromagneti(^waves.  Lord  Kel- 
vin raised  the  objection  that  if  the  sun  were  sending  out  these 
waves  in  every  direction,  it  would  give  out  as  much  energy 
in  8  hours  of  an  ordinary  electric  storm  as  it  radiates  in  light 


Fig.  155.  —  Curves  of  magnetic  storms,  prominences,  faculae,  and  sun  spotd 
from  1882  to  1904. 


and  heat  in  4  months.  A  recent  exhaustive  discussion  of  the 
data  has  led  Maunder  to  the  conclusion  that  the  source  of 
the  periodic  magnetic  storms  is  in  the  sun,  that  the  magnetic 
disturbances  are  confined  to  restricted  areas  on  the  sun,  and 
that  their  influences  are  propagated  out  from  the  sun  in 
cones  which  rotate  with  the  sun ;  that  when  these  cones  of 
magnetic  disturbances  strike  the  earth,  magnetic  storms  are 


406      AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xi,  239 

induced,  and  that  these  magnetic  storms  have  intimate, 
though  imknown,  relations  with  sun  spots.  The  most 
important  contribution  of  this  investigation  was  that  there 
is  much  observational  evidence  to  show  that  the  sun  is  not 
to  be  regarded  as  surrounded  by  a  polarized  magnetic  sphere, 
but  that  there  are  defimte  and  intense  stream  lines  of  mag- 
netic influence,  probably  connected  with  the  coronal  rays, 
reaching  out  principally  from  the  spot  zones  in  directions 
which  are  not  necessarily  exactly  radial.  It  is  a  Uttle  too 
early  to  formulate  a  precise  theory  as  to  whether  these  streams 
are  electrified  particles  driven  off  by  magnetic  forces  and 
light  pressure,  or  whether  they  involve  the  minute  corpuscles 
of  which  atoms  are  composed,  or  whether  they  are  phenomena 
of  matter  and  energy  of  a  character  and  in  a  state  not  yet 
recognized  by  science. 

XVII.    QUESTIONS 

1 .  The  apparent  diameter  of  the  sun  as  seen  from  the  earth  is  about 
32' ;  what  are  the  apparent  thicknesses  of  the  corona,  chromosphere, 
and  reversing  layer  ? 

2.  The  sun's  disk  is  considerably  brighter  at  its  center  than  near 
its  margin  (Fig.  141) :  can  this  phenomenon  be  explained  by  the  ab- 
sorption of  light  by  tne  reversing  layer  ?  By  small  solid  or  liquid 
particles  somewhere  above  the  photosphere? 

3.  If  the  smallest  spot  that  can  be  seen  subtends  an  angle  of  1', 
what  is  the  diameter  of  the  smallest  sun  spot  that  can  be  seen  simply 
through  a  smoked  glass  ? 

4.  In  what  direction  do  sun  spots  appear  to  cross  the  sun's  disk 
as  a  consequence  of  its  rotation  ? 

5.  Why  cannot  the  corona  be  observed  with  the  aid  of  the  spec- 
troscope at  any  time,  just  as  the  prominences  are  observed? 


CHAPTER  XII 
EVOLUTION   OF   THE    SOLAR    SYSTEM 

I.   General  Considerations  on  Evolution 

240.  The  Essence  of  the  Doctrine  of  Evolution.  —  The 

fundamental  basis  on  which  science  rests  is  the  orderliness 
of  the  universe.  That  it  is  not  a  chaos  has  been  confirmed 
by  an  enormous  amount  of  experience,  and  the  principle 
that  it  is  orderly  is  now  universally  accepted.  This  principle 
is  completed  in  a  fundamental  respect  by  the  doctrine  of 
evolution. 

According  to  the  fundamental  principle  of  science  the 
universe  was  orderly  yesterday,  is  orderly  to-day,  and  will 
be  orderly  to-morrow;  according  to  the  doctrine  of  evolu- 
tion, the  order  of  yesterday  changed  into  that  of  to-day  in 
a  continuous  and  lawful  manner,  and  the  order  of  to-day 
will  go  over  into  that  of  to-morrow  continuously  and  sys- 
tematically. That  is,  the  universe  is  not  only  systematic 
and  orderly  in  space,  but  also  in  time.  The  real  essence  of 
the  doctrine  of  evolution  is  that  it  maintains  the  orderliness 
of  the  universe  in  time  as  well  as  in  space. 

Evolution  may  be  from  the  simple  and  relatively  unorgan- 
ized to  the  complex  and  highly  organized,  or  it  may  be  in 
the  opposite  direction.  In  fact,  evolution  generally  involves 
the  two  types  of  changes.  For  example,  the  minerals  of  the 
soil  and  the  elements  of  the  atmosphere  sometimes  combine 
and  produce  a  tree  having  foliage,  flowers,  and  fruit.  But 
the  tree  grows,  at  least  partly,  on  the  disintegrating  products 
of  other  trees  or  plants,  and  in  its  own  trunk  the  processes 
of  decay  are  active.  Or,  to  take  a  less  commonplace  example, 
with  the  advancement  of   civihzation  men  have   become 

407 


408     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  240 

more  sensitive  to  discords  and  more  and  more  capable  of 
appreciating  certain  types  of  harmony.  There  is  almost 
certainly  a  corresponding  improvement  in  the  structure  of 
their  nervous  system.  On  the  other  hand,  there  is  degener- 
ation in  the  quaUty  of  their  teeth  and  hair.  The  changes 
in  the  two  directions  are  both  examples  of  evolution. 

As  knowledge  increases  it  is  found  that  everything  is  con- 
tinually changing.  Individuals  change,  institutions  change, 
languages  change,  and  even  the  "  eternal  hills  "  are  broken 
up  and  washed  away  by  the  elements  in  a  moment  of  geo- 
logical time.  Moreover,  all  these  changes  are  found  to  be 
perfectly  orderly.  The  doctrine  of  evolution,  as  defined  here, 
is  so  fundamentally  sensible  and  is  confirmed  by  so  much 
experience  that  scientists,  the  world  over,  accept  it  with  ab- 
solute confidence.  There  have  been,  and  there  doubtless 
will  continue  to  be,  differences  of  opinion  regarding  what 
the  precise  processes  of  certain  particular  evolutions  may 
have  been,  but  there  is  no  disagreement  whatever  regarding 
the  fundamental  principles. 

241.  The  Value  of  a  Theory  of  Evolution.  —  The  impor- 
tance of  a  general  principle  is  proportional  to  the  number 
of  known  facts  it  correlates.  This  is  a  general  proposition 
with  special  applications  in  science.  Since  a  theory  of 
evolution  is  concerned  largely  v/ith  the  relations  among 
the  data  established  by  experience,  it  naturally  forces  an 
attempt  at  their  correlation.  Moreover,  the  relations  are 
examined  in  a  critical  spirit,  so  that  any  errors  in  the  data 
or  misconceptions  regarding  their  relations  are  apt  to  be 
revealed.  Therefore,  an  attempt  to  construct  a  theory  of 
evolution  is  of  value  because  it  leads  to  a  better  understand- 
ing of  the  material  upon  which  it  is  being  based. 

A  theory  of  evolution  invariably  demands  a  knowledge 
of  facts  in  addition  to  those  upon  which  it  is  based.  In  this 
way  it  stimulates  and  directs  investigation.  A  great  major? 
ity  of  the  investigations  which  scientific  men  make  are  for 
the  purpose  of  proving  or  disproving  some  theory  they  have 


CH.  xn,  241]   EVOLUTION   OF  THE   SOLAR  SYSTEM     409 

tentatively  formulated.  The  true  scientist  often  has  pre- 
conceived notions  as  to  what  is  true,  but  he  conscientiously 
follows  the  results  of  experience. 

A  broad  scientific  theory  involves  many  secondary  theories 
depending  upon  special  groups  of  phenomena.  For  example, 
a  theory  of  the  origin  and  development  of  the  solar  system 
will  involve  theories  of  the  sun's  heat,  of  the  revolution  of 
the  planets,  of  the  rotation  of  the  planets,  of  the  planetoids, 
of  the  zodiacal  light,  etc.  In  the  construction  of  a  general 
theory  of  evolution  the  secondary  theories  are  related  to 
the  whole,  and  in  this  way  they  are  subjected  to  a  searching 
examination.  The  criticism  of  secondary  theories,  whether 
the  result  is  favorable  or  adverse,  constitutes  another  impor- 
tant value  of  the  development  of  a  theory  of  evolution. 

The  activities  of  men  are  largely  directed  toward  satisfying 
their  intellectual  wants,  though  this  fact  might  be  easily 
overlooked.  For  example,  they  do  not  ordinarily  visit  for- 
eign countries  to  get  more  to  eat  or  wear,  but  to  acquire 
broader  views  of  the  world.  The  important  thing  in  travel- 
ing is  not  that  a  person  goes  physically  to  any  particular 
place,  but  that  he  gets '  the  intellectual  experiences  that 
result  from  going  there.  Astronomers  cannot  travel  through 
the  vast  regions  of  space  which  they  explore,  but  the  long 
arms  of  their  analysis  reach  out  and  gather  up  the  facts 
and  bring  them  to  their  consciousness  with  a  vividness 
scarcely  surpassed  in  any  experience.  As  their  powerful 
instruments  and  mathematical  processes  extend  their  experi- 
ence in  space,  so  a  theory  of  evolution,  to  the  extent  that  it  is 
complete  and  sound,  extends  their  experience  in  time. 

Finally,  a  theory  which  gives  unity  to  a  great  variety  of 
observational  data  is  of  rare  aesthetic  value.  It  is  related 
to  the  catalogue  of  imperfectly  correlated  facts  upon  which 
it  is  based  as  a  finished  and  magnificent  cathedral  is  to  the 
unsightly  heaps  of  stone,  brick,  and  wood  from  which  it 
was  built.  In  some  reflections  along  this  hne,  near  the 
close  of  his   popular  work  on   astronomy,   Laplace    said, 


410     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  241 

"  Contemplated  as  one  grand  whole,  astronomy  is  the  most 
beautiful  monument  of  the  human  mind,  the  noblest  record 
of  its  intelhgence." 

In  view  of  these  considerations  it  is  evident  that  the  evolu- 
tion of  the  solar  system  is  a  subject  to  which  the  astronomer 
naturally  gives  serious  attention.  The  foremost  authorities 
of  the  present  time  have  treated  the  question  in  lectures, 
in  essays,  and  in  books.  When  new  discoveries  are  made 
their  bearings  on  evolutionary  theories  are  at  once  examined. 
Astronomers  are  rapidly  approaching  the  point  of  view  of  the 
biologists,  who  interpret  all  of  their  phenomena  in  terms  of 
evolutionary  doctrines.  Yet  scarcely  a  generation  ago  many 
astronomers  regarded  the  consideration  of  the  evolution  of 
the  solar  system  as  a  dangerous  speculation. 

242.  Outline  of  the  Growth  of  the  Doctrine  of  Evolution. 
—  Every  great  discovery  doubtless  has  been  the  culmination 
of  a  long  period  of  preUminary  work,  and  before  final  success 
has  been  attained  generally  many  men  have. approximated 
to  the  truth.  So  it  has  been  with  the  doctrine  of  evolution. 
The  ancient  Greeks  developed  theories  that  everything  had 
evolved  from  fire,  or  from  air  and  water.  These  theories 
contained  the  germ  of  the  idea  of  evolution,  but  their  authors 
had  not  laid  securely  enough  the  foundations  of  science  to 
enable  them  to  treat  successfully  the  development  of  the 
universe.  After  the  decUne  of  their  intellectual  activity 
the  subject  of  evolution  was  not  considered  seriously  for 
many  centuries. 

In  the  eighteenth  century  geologists  were  groping  for  a 
satisfactory  theory  regarding  the  succession  of  the  Ufe 
forms  whose  fossils  were  found  in  the  rocks.  They  seem  to 
have  concluded  on  the  whole  that  the  earth  had  been  sub- 
ject to  a  number  of  great  cataclysms  in  which  all  Hfe  was 
destroyed.  They  supposed  that  following  each  destruction  of 
Ufe  there  had  been  a  new  creation  in  which  higher  forms  were 
produced.  The  prevalence  of  such  ideas  as  these  shows  with 
what  difficulty  the  doctrine  of  evolution  was  developed. 


OH.  XII,  }i^}   jBivUljUTlON  OF  THE   SOLAR  SYSTEM     411 

In  1750  Thomas  Wright,  of  Durham,  England,  published 
a  theory  of  the  evolution,  not  only  of  the  solar  system,  but 
also  of  all  the  stars  that  fill  the  sky.  The  chief  merit  of 
this  work  was  that  indirectly  it  gave  a  straightforward 
exposition  of  the  doctrine  of  evolution.  Its  chief  influence 
seems  to  have  been  on  the  young  philosopher  Kant,  into 
whose  hands  it  fell.  Kant  at  once  turned  his  briUiant  mind 
to  the  contemplation  of  the  problems  of  cosmogony,  or  the 
evolution  of  the  celestial  bodies,  and  in  1755  he  published 
a  remarkable  book  on  the  subject.  But  the  world  seems  not 
to  have  been  ripe  for  the  idea  of  evolution,  because  neither 
the  work  of  Wright  nor  that  of  Kant  had  any  important 
influence  upon  science. 

In  1796  the  great  French  astronomer  and  mathematician 
Laplace  published  his  celebrated  "  Nebular  Hypothesis."  It 
was  supported  by  the  great  name  of  its  author,  and  it  was 
relatively  simple  and  easily  understood.  Moreover,  during 
the  French  Revolution  the  world  had  acquired  a  new  point 
of  view  and  had  become  more  receptive  of  new  ideas.  For 
these  reasons  the  theory  of  Laplace  soon  obtained  wide 
acceptance  among  scientific  men.  It  made  a  profound 
impression  on  geologists  because  it  furnished  them  with  an 
account  of  the  early  history  of  the  earth.  It  gave  them 
astronomical  authority  for  an  originally  hot  and  molten 
earth  which  had  soUdified  on  cooUng.  It  encouraged  them 
to  interpret  geological  phenomena  by  geological  principles. 
In  the  early  decades  of  the  nineteenth  century  geologists 
largely  abandoned  the  idea  that  the  earth  had  necessarily 
been  visited  by  destructive  cataclysms,  and  adopted  the  view 
that  it  had  undergone  a  continuous  series  of  great  changes 
at  a  roughly  uniform  rate. 

The  work  of  the  geologists  led  naturally  to  the  extension 
of  the  doctrine  of  evolution  to  the  biological  sciences.  In 
the  first  place,  the  belief  that  the  earth  was  enormously 
old  had  become  current.  In  the  second  place,  there  were 
unmistakable  evidences  that  the  surface  of  the  earth  had 


412     AN   INTRODUCTION   TO  ASTRONOMY    ICH.  xii,  ^^ 

undergone  extensive  changes.  In  the  third  place,  the  early 
rocks  contained  only  fossils  of  low  forms  of  Ufe,  while  the 
later  rocks  contained  fossils  of  higher  forms  of  hfe.  In  addi- 
tion, there  were  many  direct  evidences  of  a  purely  biological 
character  that  there  was  an  almost  continuous  series  of  Ufe 
forms  from  the  lowest  to  the  highest. 

The  principle  of  biological  evolution  seems  to  have  been 
taking  definite  shape  simultaneously  in  the  minds  of  Charles 
Darwin,  Alfred  Russel  Wallace,  and  Herbert  Spencer. 
Darwin  and  Wallace  were  naturalists  and  Spencer  was  a 
philosopher.  In  1858  Darwin  pubhshed  his  Origin  of 
Species,  in  which  he  brought  together  the  results  of  almost 
a  hfetime  of  keen  observations  and  profound  reflections. 
He  gave  unanswerable  evidence  for  his  conclusion  that 
during  the  geological  ages,  as  a  consequence  of  changing 
environment,  natural  selection,  survival  of  the  fittest,  etc., 
one  species  of  animals  gradually  changed  into  another,  and 
that  at  the  present  time  all  the  higher  types  of  animals, 
including  man,  are  more  or  less  closely  related. 

In  spite  of  the  fact  that  the  doctrine  of  evolution  is  full  of 
hope  for  the  future  progress  of  the  human  race,  Darwin's 
book  aroused  the  bitterest  antagonism.  While  biologists  do 
not  now  fully  agree  with  him  as  to  the  relative  importance 
of  the  various  factors  involved  in  biological  evolution,  never- 
theless they  universally  accept  his  fundamental  conclusions. 
Moreover,  the  changes  in  political,  social,  and  rehgious  insti- 
tutions are  now  considered  in  the  Ught  of  the  same  ideas. 
That  is,  the  condition  of  the  whole  universe  at  one  time 
evolves  continuously  and  in  obedience  to  all  the  factors  op- 
erating on  it  into  that  which  exists  at  another  time. 

In  brief,  the  development  of  the  modern  doctrine  of  evolu- 
tion is  as  follows :  In  the  middle  of  the  eighteenth  century 
its  first  beginnings  were  laid  in  astronomy  by  Wright  and 
Kant.  At  the  end  of  the  century  it  was  given  an  enormous 
impulse  by  the  astronomer  and  mathematician,  Laplace. 
His  theory  of  the  origin  of  the  earth  stimulated  geologists 


CH.  XII,  243]   EVOLUTION  OF  THE   SOLAR  SYSTEM     413 

to  adopt  it  in  the  early  decades  of  the  nineteenth  century. 
By  the  middle  of  the  century  it  was  being  definitely  applied 
in  the  biological  sciences.  In  1858  Darwin  published  his 
great  masterpiece,  The  Origin  of  Species,  which  gave  the 
whole  world  a  new  point  of  view  and  revolutionized  its 
methods  of  thought.  The  development  and  adoption  of  the 
doctrine  of  evolution  was  the  greatest  achievement  of  the 
nineteenth  century. 

XVIII.     QUESTIONS 

1.  Is  the  erosion  of  the  chasm  below  Niagara  Falls  an  example 
of  an  evolution?  Is  the  clearing  away  of  the  forests  and  the  prep- 
aration of  the  land  for  cultivation  ?     Is  an  explosion  of  dynamite  ? 

2.  Would  the  direct  creation  of  men  and  lower  animals  be  an 
example  of  evolution? 

3.  Do  the  changes  in  scientific  ideas  constitute  an  evolution? 

4.  Are  religious  ideas  undergoing  an  evolution? 

5.  Will  the  doctrine  of  evolution  undergo  an  evolution? 

6.  The  universe  in  our  vicinity  at  the  present  time  is  believed 
to  be  orderly ;  is  it  reasonable  to  suppose  that  in  remote  regions  or 
at  remote  times  it  was  not  orderly  ? 

7.  Why  was  the  doctrine  of  evolution  first  clearly  understood  in 
astronomy  ? 

8.  According  to  the  doctrine  of  evolution,  will  two  identical 
conditions  of  the  universe  lead  to  identical  results  ?  Is  it  probable 
that  the  universe  is  twice  in  exactly  the  same  state  ? 

II.  Data  of  the  Problem  of  Evolution  of  the  Solar 

System 

243.   General    Evidences    of    Orderly    Development.  — 

There  are  certain  obvious  evidences  that  the  solar  system 
has  undergone  an  orderly  evolution.  For  example,  the 
planets  all  revolve  around  the  sun  in  nearly  the  same  plane 
and  in  the  same  direction.  There  are  in  addition  over  800 
planetoids  which  have  similar  motions.  Moreover,  the  sun 
and  the  four  planets  whose  surface  markings  are  distinctly 
visible  rotate  in  the  same  direction.  So  great  a  uniformity 
can  scarcely  be  the  result  of  chance. 


414     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  ^»3 

In  order  to  treat  the  matter  numerically,  suppose  there  are 
800  bodies  whose  planes  of  motion  do  not  differ  from  the 
plane  of  the  earth's  orbit  by  more  than  18°,  and  whose 
directions  of  motion  are  the  same  as  that  of  the  earth.  Since 
the  incUnation  of  an  orbit  could  be  anything  from  0°  to  180°, 
the  chance  that  it  would  lie  between  0°  and  18°  is  ■^.  The 
probability  that  the  planes  of  the  orbits  of  two  bodies  would 
be  less  than  18°  is  (t^)^  And  the  probability  that  the 
same  would  be  true  for  800  bodies  is  only  (yV)  *"",  or  unity 
divided  by  1  followed  by  800  ciphers.  This  probability  is 
so  small  that  we  are  forced  to  the  conclusion  that  the  arrange- 
ment of  the  planets  in  the  solar  system  iS  not  accidental. 
Both  Kant  and  Laplace  made  use  of  this  line  of  reasoning. 

A  planet  may  revolve  around  the  sun  in  an  orbit  of  any 
eccentricity  from  0  to  1.  Of  the  more  than  800  planets 
and  planetoids,  the  orbits  of  624  have  eccentricities  less  than 
0.2,  the  orbits  of  all  except  26  have  eccentricities  less  than 
0.3,  and  the  orbit  of  only  one  has  an  eccentricity  greater 
than  0.5.  These  remarkable  facts  imply  that  some  sys- 
tematic cause  has  been  at  work  which  has  produced  plan- 
etary orbits  of  low  eccentricity.  And  both  the  positions  of 
the  planes  and  the  small  eccentricities  of  the  orbits  of  the 
planets  prove  conclusively  that  the  solar  system,  in  all  its 
history,  has  not  been  subject  to  any  important  external  dis- 
turbance, such  as  a  closely  passing  star. 

244.  Distribution  of  Mass  in  the  Solar  System.  —  Nearly 
all  the  matter  of  the  solar  system  is  concentrated  in  the  sun. 
In  fact,  all  the  planets  together  contain  less  than  one  seventh 
of  one  per  cent  of  the  mass  of  the  entire  system.  Although 
the  mass  of  Jupiter  is  more  than  2.5  times  that  of  all  the 
other  planets  combined,  it  is  less  than  one  thousandth  that 
of  the  sun. 

It  is  important  to  know  whether  the  masses  of  the  sun 
and  planets  are  now  changing.  There  is  certainly  at  pres- 
ent no  appreciable  transfer  of  matter  from  one  body  to 
another.    The  sun  may  be  losing  some  particles  by  ejecting 


CH.  XII,  244]   EVOLUTION   OF  THE   SOLAR  SYSTEM     415 

them  from  its  surface  in  an  electrified  condition,  and  a  very- 
small  percentage  of  the  ejected  particles  may  strike  the 
planets,  but  it  is  very  improbable  that  the  process  has  had 
important  effects  on  the  distribution  of  mass  in  the  solar 
system,  even  in  the  enormous  intervals  of  time  required  for 
its  evolution. 

The  mass  of  the  earth  is  slowly  increasing  by  the  meteoric 
material  which  it  sweeps  up  in  its  journey  around  the  sun. 
It  is  not  unreasonable  to  suppose  that  the  other  planets, 
and  possibly  the  sun,  are  growing  similarly.  This  growth, 
at  least  in  the  case  of  the  earth,  is  too  slow  at  present  to 
have  a  very  important  bearing  on  the  evolution  of  the 
whole  system.  But  if  the  meteors  are  permanent  members 
of  the  solar  system,  the  more  they  are  swept  up  by  the 
planets  the  more  infrequent  they  become  and  the  smaller  the 
number  a  planet  encounters  in  a  day.  Consequently,  the 
acquisition  of  meteoric  niaterial  by  collision  may  once  have 
been  a  much  more  important  factor  in  the  evolution  of 
the  planets  than  it  is  at  the  present  time.  In  fact,  so  far 
as  general  considerations  go,  appreciable  fractions  of  the 
masses  of  the  planets  may  have  been  obtained  from  meteoric 
material.  But  it  is  improbstble  that  the  great  sun  has 
grown  sensibly  in  this  way. 

It  follows  from  this  discussion  that  probably  the  remote 
antecedent  of  the  solar  system  consisted  of  an  overwhelm- 
ing central  mass  and  a  very  small  quantity  of  matter  dis- 
tributed somewhat  irregularly  out  from  it  to  an  enormous 
distance.  At  any  rate,  if  this  were  not  the  original  distribu- 
tion of  matter,  the  conditions  must  have  been  such  that  the 
central  condensation  resulted  in  harmony  with  the  laws 
of  dynamics.  The  ever-increasing  distances  between  the 
planets  is  shown  in  Figs.  96  and  97.  The  relatively  small 
masses  of  the  planets  and  their  enormous  distances  from  one 
another  are  among  the  most  remarkable  facts  that  need  to 
be  taken  into  account  when  considering  their  origin  and 
evolution. 


416     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  244 

An  additional  fact  which  must  be  noted  is  that  the  ter- 
restrial planets  contain  the  heaviest  known  substances.  The 
sun  also  contains  heavy  elements  (Art.  234),  though  the 
spectral  lines  of  the  very  heaviest  have  not  been  found. 
The  constitution  of  the  large  planets  is  not  so  well  known, 
though  it  may  be  inferred  from  their  low  densities  and  mod- 
erate temperatures  that  they  contain  largely  only  the  Kght 
elements.  Any  hypothesis  as  to  the  origin  of  the  planets, 
in  order  to  be  satisfactory,  must  make  provision  for  this 
distribution  of  the  elements. 

245.  Distribution  of  Moment  of  Momentum.  —  In  at- 
tempting to  go  back  to  the  origin  of  the  solar  system"  it  is 
natural  to  consider  its  mass  and  distribution  of  mass  because 
matter  is  indestructible.  For  a  similar  reason,  the  distribu- 
tion of  the, moment  of  momentum  of  the  system  among  its 
various  members  is  of  fundamental  importance.  That  is, 
if  the  solar  system  has  undergone  its  evolution  free  from 
exterior  disturbances,  its  total  moment  of  momentum  is 
now  exactly  equal  to  what  it  was  at  the  beginning  and  at 
every  stage  of  its  development. 

As  has  been  stated,  the  small  mutual  inclinations  of  the 
orbits  of  the  planets  and  the  small  eccentricities  ol  their 
orbits  both  prove  that  the  solar  system  has  been  subj  ect 
to  no  important  exterior  influences  since  the  planets  were 
formed.  Hence  any  hypothetical  antecedent  of  the  system 
must  be  assigned  the  quantity  of  moment  of  momentum  it 
now  possesses.  Although  this  fact  is  perfectly  clear,  it  was 
overlooked  by  Kant  and  was  not  given  adequate  consider- 
ation by  Laplace  and  his  followers. 

In  Table  XII  the  mass  and  moment  of  momentum  is 
given  for  the  sun  and  each  of  the  eight  planets  in  such  units 
that  the  sums  are  unity.  The  moment  of  momentum  of  the 
sun  depends  upon  its  law  of  density.  In  the  computation  it 
was  assumed  that  the  mass  is  concentrated  toward  the  in- 
terior according  to  a  law  of  increase  of  density  formulated 
by  Laplace.     The  rotations  of  the  planets  contribute  so 


CH.  XII,  245]   EVOLUTION  OP   THE   SOLAR   SYSTEM     417 


little  to  the  final  results  that  it  is  not  important  what  law  of 
density  is  used  for  them. 


TABL33    XII 


Body 

Mass 

Moment  of 
Momentum 

Sun 

Mercury  .          ... 
Venus  ...          ... 

Earfh        

Mars         .     .          ... 
Jupiter          .     .     . 
Saturn            .     . 
Uranus          .     . 
Neptune  .... 

0.9986590 
0.0000001 
0.0000025 
0.0000030 
0.0000003 
0.0009558 
0.0002852 
0.0000430 
0.0000511 

0.027423 

0.000017 

0.000576 

0.000827 

0.000112 

0.599273- 

0.241924 

0.052845 

0.077003 

Total    . 

1.0000000 

1.000000 

It  is  seen  from  this  table  that  although  the  mass  of  the  sun 
is  700  times  as  great  as  that  of  all  the  planets  combined, 
its  moment  of  momentum  is  only  a  httle  over  -^  that  of  the 
planets.  Or,  considering  the  material  interior  to  the  orbit 
of  Saturn,  it  is  found  that  while  Jupiter  contains  only  -^ 
of  one  per  cent,  or  xwmt  o^  t^®  entire  mass,  it  possesses 
more  than  95  per  cent  of  the  moment  of  momentum. 

One  at  once  inquires  whether  the  distribution  of  moment 
of  momentum  is  now  being  changed.  The  mutual  attrac- 
tions of  the  planets  produce  some  changes  in  the  distribu- 
tion of  moment  of  momentum,  but  they  are  of  no  importance 
whatever  in  connection  with  the  problem  under  consideration. 
The  tides  which  a  planet  generates  in  the  sun  reduce  the 
moment  of  momentum  of  the  sun  and  increase  that  of  the 
ptanet.  But  here  again  the  results  are  inappreciable  even 
for  thousands  of  milUons  of  years.  The  earth  encounters 
meteoric  matter  in  its  revolution  around  the  sun,  and  it  is 
probable  that  the  other  planets  are  subject  to  similar  dis- 
tmbances.-  The  result  of  the  resistance  by  meteors  is  to 
reduce  the  moment  of  momentum  oYthe  planets^-  -At  pres- 
2e 


418     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  245 

ent  the  effects  of  meteors  on  the  motion  of  the  earth  are 
inappreciable,  but  it  is  not  certain  that  they  were  not  once 
important.  However,  whether  or  not  they  have  ever  been 
of  importance,  they  cannot  relieve  the  inequalities  in  the 
table,  for  they  are  decreasing  the  moment  of  momentum  of 
the  planets,  which  are  still  relatively  very  large.  In  fact, 
there  have  been  no  known  influences  at  work  which  could 
have  sensibly  modified  the  distribution  of  the  moment  of 
momentum  of  the  system  since  the  sun  and  planets  have 
been  separate  bodies. 

It  remains  to  inquire  whether  the  sun  and  planets  may  not 
once  have  been  parts  of  one  mass  with  a  distribution  of 
moment  of  momentum  quite  different  from  that  found  at 
present.  Since  the  planets  are  not  receding  from  the  sun, 
the  only  possibility  is  that  the  sun  and  the  planets  Were 
formerly  so  expanded  that  the  material  of  which  they  are 
composed  was  more  or  less  intermingled. 

According  to  the  contraction  theory  of  the  heat  of  the  sun, 
the  sun's  dimensions  were  formerly  greater  than  they  are 
at  present.  Indeed,  the  sun  has  been  supposed  to  have 
once  filled  all  the  space  now  occupied  by  the  planets.  Fol- 
lowed backward  in  time,  the  sun  is  found  to  be  larger  and 
larger,  rotating  more  and  more  slowly  because  its  moment 
of  momentum  remained  constant  during  contraction,  and 
more  and  more  nearly  spherical  because  a  rotating  body 
becomes  more  oblate  with  contraction.  It  follows  from  the 
table  that  if  the  planets  which  are  interior  to  Jupiter  were 
added  to  the  sun  they  would  not  have  an  important  effect 
on  its  moment  of  momentum. 

Now  suppose  the  sun  was  once  expanded  out  to  the  orbit 
of  Jupiter ;  its  radius  was  more  than  1000  times  its  present 
radius,  its  volume  was  more  than  1000*  =  1,000,000,000 
times  its  present  volume,  and  its  density  was  correspond- 
ingly less.  Even  if  it  was  not  condensed  toward  the  center, 
the  density  at  its  periphery  was  then  less  than  one  millionth 
of  that  of  the  earth's  atmosphere  at  sea  level.     It  follows  from 


CH.  XII,  246]   EVOLUTION  OF  THE   SOLAR  SYSTEM     419 

the  fact  that  the  moment  of  momentum  was  necessarily- 
constant,  that  its  period  of  rotation  must  have  been  about 
70,000  years.  But  Jupiter's  period  of  revolution  is  about 
12  years.  Now,  therefore,  either  Jupiter  was  then  quite 
independent  of  the  general  solar  mass ;  or,  if  not,  in  some 
unknown  way  this  extremely  tenuous  material  must  have 
imparted  to  that  minute  fraction  of  itself  which  later  became 
Jupiter  enough  moment  of  momentum  to  reduce  the  period 
of  this  part  from  70,000  years  to  12  years.  More  specifically, 
it  is  seen  from  the  table  that  Jupiter,  which  contains  one 
tenth  of  one  per  cent  of  the  mass  of  the  solar  system  within 
the  orbit  of  Saturn,  carries  over  95  per  cent  of  the  moment 
of  momentum.  It  is  incredible  that  this  extreme  distribu- 
tion of  moment  of  momentum  could  have  developed  from  an 
approximately  uniform  distribution,  especially  in  a  mass 
of  such  low  density,  and  no  one  has  been  able  to  formulate 
a  plausible  explanation  of  it.  Consequently,  it  must  be 
concluded  that  the » distribution  of  moment  of  momentum 
in  the  solar  system  has  not  changed  appreciably  since  it  has 
been  free  from  important  exterior  forces. 

246.  The  Energy  of  the  Solar  System.  —  In  considering 
the  energy  of  the  solar  system,  the  discussion  must  include 
its  kinetic  energy,  heat  energy,  potential  energy,  and  sub- 
atomic energy. 

The  kinetic  energy  of  a  body  is  its  energy  of  motion 
including  translation,  rotation,  and  internal  currents.  The 
kinetic  energy  of  the  solar  system  consists  of  its  energy  of 
translation  and  of  the  internal  motions  of  its  parts.  The 
former  cannot  have  changed  except  by  the  action  of  exterior 
forces.  Moreover,  its  value  is  not  accurately  known,  and 
it  has  no  relation  to  the  remaining  energy  of  the  system  so 
long  as  no  other  celestial  body  is  encountered.  Therefore 
it  will  be  given  no  further  consideration  in  this  connection. 
The  mutual  attractions  of  the  planets  change  their  transla- 
tory  motions,  but  in  such  a  way  that  the  sum  of  their  kinetic 
and  potential  energies  remains  constant. 


420     AN   INTRODUCTION   TO   ASTRONOMY   ,[ch.  xii,  246 

The  sun,  planets,  and  satellites  raise  tides  in  one  another. 
In  these  tides  there  is  some  friction  in  which  kinetic  energy 
degenerates  into  heat  energy,  which  is  radiated  away  into 
space.  In  this  way  the  solar  system  is  losing  energy.  The 
heat  energy  from  all  other  sources  is  likewise  being  lost  by 
radiation. 

The  potential  energy  of  a  system  is  equal  to  the  work 
which  may  be  done  upon  it,  in  virtue  of  the  relative  positions 
of  its  parts,  by  the  forces  to  which  it  is  subject.  For  example, 
a  body  100  feet  above  the  surface  of  the  earth  is  subject  to 
the  attraction  of  the  earth.  The  earth  would  do  a  certain 
amount  of  work  upon  the  body  in  causing  it  to  faU  from  an 
altitude  of  100  feet  to  its  surface.  This  work  equals  the 
potential  energy  of  the  body  in  its  original  position.  In 
the  case  of  the  translations  of  the  planets,  as  has  been  stated, 
the  sum  of  their  kinetic  and  potential  energies  is  constant. 
But  if  the  sun  or  a  planet  contracts,  the  potential  energy  of 
its  expanded  condition  is  transformed  into  heat  (Art.  216), 
which  is  at  least  partly  lost  by  radiation.  In  this  way  the 
total  energy  of  the  system  decreases,  and  the  diminution  may 
be  large  in  amount. 

There  is  certainly  a  large  amount  of  subatomic  energy  in 
uranium,  radium,  and  probably  in  all  other  elements.  In  the 
case  of  the  radioactive  substances  this  energy  is  slowly  trans- 
formed into  heat,  which  is  dissipated  by  radiation.  As  has 
been  suggested  (Art.  219),  the  subatomic  ener^es  may  be 
liberated  in  great  quantities  under  the  extreme  conditions 
of  pressure  and  temperature  which  prevail  in  the  interior 
of  the  sun. 

Since  the  solar  system  is  losing  energy  in  several  ways  and 
acquiring  only  inappreciable  amounts  from  the  outside,  as, 
for  example,  the  radiant  energy  received  from  the  stars,  it 
originally  had  more  energy  than  at  present,  and  this  condi- 
tion must  be  satisfied  by  all  hypotheses  respecting  its 
evolution. 


CH.  XII,  k!4yj   JiVULiUTlON  OF   THE   SOLAR   SYSTEM     421 

XIX.    QUESTIONS 

1.  What  is  the  probability  that  when  3  coins  are  tossed  up  they 
win  aU  fall  heads  up  ?  What  is  the  probability  that  in  a  throw  of 
4  dice  there  will  be  4  aces  up  ?  If  100  coins  were  found  heads  up, 
could  it  reasonably  be  supposed  that  the  arrangement  was  acci- 
dental? How  would  its  probability  compare  with  that  that  the 
positions  of  the  orbits  of  the  planets  and  planetoids  are  accidental  ? 

2.  Suppose  a  star  should  pass  near  the  solar  system  in  the  plane 
of  the  orbits  of  the  planets ;  would  it  disturb  the  positions  of  the 
planes,  or  the  eccentricities,  of  their  orbits  ? 

3.  How  many  tons  of  meteors  would  have  to  strike  the  earth 
.  daily  in  order  to  double  its  mass  in  200,000,000  years  ?    How  many 

would  daily  strike  each  square  mile  of  its  surface  ? 

4.  What  is  the  definition  of  moment  of  momentum?  How 
does  it  differ  from  momentum  ?  Is  it  manifested  in  various  forms 
hke  energy  ?  Does  the  loss  of  energy  of  a  body  by  radiation  change 
its^ moment  of  momentum? 

5.  The  mass  of  the  earth  is  1.2  times  that  of  Venus  (Table  XII) ; 
why  is  its  moment  of  momentum  more  than  1.2  times  that  of 
Venus  ? 

6.  Could  the  total  energy  of  the  solar  system  have  been  infinite 
at  the  start?  Can  the  system  have  existed  in  approximately  its 
present  condition  for  an  infinite  time  ? 

7.  When  carbon  and  oxygen  unite  chemically,  heat  is  produced ; 
is  this  heat  energy  developed  at  the  expense  of  the  kinetic,  potential, 
heat,  or  subatomic  energies  of  the  original  materials  ? 

III.   The  Planetesimal  Hypothesis^ 

247.  Brief  Outline  of  the  Planetesimal  Hypothesis. — 

The  fundamental  cojiditions  imposed  by  the  distribution  of 
mass  and  moment  of  momentum  in  the  solar  system,  together 
with  many  supplementary  considerations,  have  led  to  the 
planetesimal  hypothesis.  According  to  this  hypothesis,  the 
remote  ancestor  of  the  solar  system  was  a  more  or  less  con- 
densed and  well-defined  central  sun,  having  slow  rotation, 
surrounded  by  a  vast  swarm  of  somewhat  irregularly  scat- 
tered  secondary  bodies,   or  planetesimals    (little  planets) , 

■  The  Planetesimal  Hypothesis  was  developed  by  Professor  T.  C.  Cham- 
berlin  and  the  author  in  1900  and  the  following  years. 


422     AN   INTRODUCTION   TO  ASTRONOMY    Ich.  xii,  m, 

which  all  revolved  in  elliptical  orbits  about  the  central  mass , 
in  the  same  general  direction.     This  organization  evidently 
satisfies  the  data  of  the  problem.     Moreover,   the  spiral 
nebulas  [Art.  302]  offer  numerous  examples  of  matter  which 
is  apparently  in  this  state. 

According  to  the  planetesimal  hypothesis,  our  present 
sun  developed  from  the  central  parent  mass  and  possibly 
some  outlying  parts  which  fell  in  upon  it  because  they  had 
small  motions  of  translation.  The  revolving  scattered  mate- 
rial  contained  nuclei  of  various  dimensions  which,  in  their 
motions  about  the  central  sun,  swept  up  the  remaining 
scattered  material  and  gradually  grew  into  planets  whose 
masses  depend  upon  the  original  masses  of  the  nuclei  and 
the  amount  of  matter  in  the  regions  through  which  thev 
passed.  The  angles  between  the  planes  of  the  orbits  were 
gradually  reduced  by  the  colhsions,  and  at  the  same  time 
the  eccentric  orbits  became  more  nearly  circular.  In  the 
processo!  growth  TEe  planetary  nuclei  acquired  their  forward 
rotations. 

248.  Examples  of  Planetesimal  Organization.  —  The' 
planetoids  afford  a  trace  of  the  former  planetesimal  condi- 
tion of  the  solar  system.  The  average  inclination  and  the 
average  eccentricity  of  their  orbits,  are  considerably  larger 
than  the  corresponding  quantities  for  the  planets.  If  the 
region  which  they  occupy  had  been  swept  by  a  dominating 
nucleus,  they  would  have  combined  with  it  in  a  planet  occupy- 
ing approximately  the  mean  position  of  the  planes  of  their 
orbits  and  having  a  small  eccentricity  (Art.  252). 

Anotjier  example  of  planetesimal  organization  is  fur- 
nished by  the  particles  of  which  the  rings  of  Saturn  are 
composed.  One  might  at  first  thought  conclude  that  they 
would  have  formed  one  or  more  satelhtes  if  dominating  nuclei 
had  been  revolving  around  the  planet  in  the  zone  which 
they  occupy.  But  they  are  very  close  to  Saturn,  and  a  satel- 
lite revolving  at  their  distance  would  be  subject  to  the  strains 
of  the  tides  produced  by  the  planet.     As  has  been  stated 


CH.  XII,  248]   EVOLUTION  OP   THE    SOLAR   SYSTEM     423 

(Art.  183),  Roche  showed  that  a  fluid  satelUte  could  not  re- 
volve withih  2.44  radii  of  a  planet  without  being  broken 
up,  unless  its  density  were  greater  than  that  of  the  planet. 
Since  the  rings  of  Saturn  are  within  this  limit,  it  follows 
that  they  could  not  have  formed  a  satellite,  and  that  a 
large  nucleus  revolving  among  them,  instead  of  sweeping 
them  up,  would  itself  have  been  reduced  to  the  planetesimal 
condition,  unless  it  was  solid  and  strong  enough  to  withstand 
great  tidal  strains. 

The  examples ,  of  planetesimal  organization  which  have 
been  given  may  not  be  very  convincing.  But  we  may  inquire 
whether  there  are  not  numerous  examples  in  the  heavens, 
beyond  the  solar  system,  confirmatory  of  the  planetesimal 
theory.  The  answer  is  in  the  affirmative.  There  are  tens 
of  thousands  of  spiral  nebulae  that  are  almost  certainly  in 
the  planetesimal  condition,  though  on  a  tremendous  „scale. 
They  consist  of  central  sunhke  nuclei  which  are  generally! 
well  defined,  and  arms  of  widespreading,  scattered  niaterial. 
Their  arms  in  most  cases  probably  contain  large  masses,  but 
they  are  small  in  comparison  with  the  central  suns.  Their 
great  numbers  iniply  that  they  are  in  general  semi-perma- 
nent in  character.  Consequently,  the  material  of  which 
the  arms  are  composed,  cannot  in  general  be  moving  along 
them,  either  in  toward  or  out  from  the  central  nucleus,  for 
under  these  circutnstances  they  would  condense  into  suns 
or  dissipate  into  space,  and  in  either  case  lose  their  peculiar 
characteristics.  Besides  this,  matter  subject  to  the  law 
of  gravitation  could  not  move  along  the  arms  of  spirals.  It 
is  therefore  beheved  that  in  a  spiral  nebula  the  arms  are 
composed  of  material  which,  instead  of  proceeding  along 
them,  moves  across  them  around  the  central  nucleus  as 
a  focus.  The  spirals  owe  their  coils  to  ithe  fact  that  the 
inner  parts  revolve  faster  than  the  outer  parts.  As  a 
rule  they  radiate  white  Ught,  which  indicates  that  they  are 
at  least  partly  in  a  solid  or  liquid  state.  When  a  spiral  is 
seen  edgewise  to  the  earth  there  is  a  dark  band  through  its 


424     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  248 

center,  doubtless  produced  by  dark,  opaque  material  revolv- 
ing at  its  periphery. 

While  a  few  spiral  nebulae  have  been  known  for  a  long 
time,  their  great  numbers  were  not  suspected  until  Keeler 
began  to  photograph  them  with  the  Crossley  reflector  at  the 
Lick  Observatory.  In  a  paper  published  in  1900  shortly 
before  his  death,  he  said : 

"1.  Many  thousands  of  unrecorded  nebulas  exist  in  the 
sky.  A  conservative  estimate  places  the  nimiber  within  the 
reach  of  the  Crossley  reflector  at  about  120,000.  The  number 
of  nebulae  in  our  catalogues  is  but  a  small  fraction  of  this. 

"  2.  These  nebulae  exhibit  all  gradations  of  apparent  size 
from  the  great  nebula  in  Andromeda  down  to  an  object 
which  is  hardly  distinguishable  from  a  faint  star  disk. 

"3.  Most  of  these  nebulae  have  a  spiral  structure.  .  .  . 
While  I  must  leave  to  others  an  estimate  of  the  importance 
of  these  conclusions,  it  seems  to  me  that  they  have  a  very 
direct  bearing  on  many,  if  not  all,  questions  concerning  the 
cosmogony.  If,  for  example,  the  spiral  is  the  form  normally 
assumed  by  a  contracting  nebulous  mass,  the  idea  at  once 
suggests  itself  that  the  solar  system  has  been  evolved  from  a 
spiral  nebula,  while  the  photographs  show  that  the  spiral 
is  not,  as  a  rule,  characterized  by  the  simplicity  attributed  to 
the  contracting  mass  in  the  nebular  (Laplacian)  hypothesis. 
This  is  a  question  which  has  already  been  taken  up  by 
Chamberlin  and  Moulton  of  the  University  of  Chicago." 

WhjlB  the  spirals  are  almost  certainlv  examples  of  plan- 
etesimal  organization,  those  which  have  been  photographed 
are  enormously  larger  than  the  parent  of  the  solar  system 
unless,  indeed,  there  are  many  undiscovered  planets  beyond 
the  orbit  of  Neptune.  But,  as  Keeler  remarked,  there  is  no 
lower  limit  to  the  apparent  dimensions  of  the  spiral  nebulae, 
and  it  is  possible  that  many  of  them  are  actually  of  very 
moderate  size. 

249.  Suggested  Origin  of  Spiral  Nebulae.  —  Although  the 
validity  of  the  planetesimal  theory  does  not  hang  upon  any 


CH.  XII,  249]   EVOLUTION  OF  THE   SOLAR   SYSTEM     425 

hypothesis  as  to  the  origin  of  spiral  nebulae,  yet,  if  the  solar 
system  has  evolved  from  a  spiral  nebula,  the  theory  of  its 
origin  will  not  be  regarded  as  complete  and  fully  satisfac- 
tory until  the  mode  of  generation  of  these  nebulae  has  been 
explained.  The  be^t  suggestion  regarding  their  genesis, 
which  is  due  primarily  to  Chamberlin,  is  as  follows : 

There  are  several  hundreds  of  milhons  of  stars  in  the 
heavens  and  they  are  moving  with  respect  to  one  another  with 
an  average  velocity  of  about  600,000,000  miles  per  year. 
While  their  motions  are  by  no  means  entirely  at  random,  yet 
there  are  milHons  of  them 
moving  in  essentially 
every  direction.  It  is  in- 
evitable  that  in  the  course 
of  time  everv  star  will  pass 
near  some  other  star.  If 
two  stars  should  collide, 
the  energv  of  their  motion 
would  largely  be  changed 
into  heat  and  the  com- 
bined  mass  would  be  trans- 


formed   into    a    gaseous 


nebula.       If    they    should    Fig.  156. — Deflection  of  ejected  material 
.  by  a  passing  star. 

simply  pass  near  one  an- 
other without  striking,  an  event  which  would  occur  many 
times  more  frequently  than  a  collision,  a  spiral  nebula  would 
probably  be  formed,  as  will  now  be  shown. 

Consider  two  stars  passing  near  each  other.  They  both 
move  about  their  center  of  gravity,  but  no  error  will  be 
committed  in  representing  one  of  them  as  being  at  rest  and 
the  other  as  passing  by  it.  If  the  stars  are  equal,  their 
effects  on  each  other  are  the  same,  but  in  order  not  to  divide 
the  attention,  only  the  action  of  S'  on  S  will  be  considered. 

Consider  S'  when  it  is  at  the  position  Si,  Fig.  156.  It 
raises  tides  on  S^  one  on  the  side  toward  S'  and  the  other  on 
the  opposite  side.    The  heights  of  the  tides  depend  upon  the 


426     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  249 


relative  masses  of  the  two  suns  and  their  distance  apart 
compared  to  the  radius  of  S.  Ap  approach  within  10,000,000 
miles  is  more  than  100  times  as  probable  as  even  a  grazing 
collision.  At  tliis  distance  the  tide-raising  force  of  S'  on  S 
compared  to  the  surface  gravity  of  S  is  more  than  2000 

times  the  tide-raising 
force  of  the  moon  on  the 
eartn  compared  to  the 
surface  gravity  of  the 
earth.  The  tide-raising 
force  varies  directly  as 
the  radius  of  the  dis- 
turbed body  and  in- 
versely as  the  cube  of  the 
distance  of  the  disturb- 
ing body  (Art.  153). 
Hence,  if  the  nearest 
approach  wjere  5,000,000 
miles,  the  tide-raising 
force  would  be  more  than 
16,000  times  greater, 
relatively  to  surface 
gravity,  than  that  of  the 
moon  on  the  earth.  This 
force  would  raise  tides 
approximately  500  miles 
high  if  the  sun  were  a 
homogeneous  fluid,  and 
there  would  be  a  corresponding  slight  constriction  of  the  sun 
in  a  belt  midway  between  the  tidal  cones.  The  tides  on  a 
highly  heated  gaseous  body  would  probably  be  much  higher. 
^  The  sun  is  the  seat  of  violent  explosive  forces  which  now 
often  eject  matter  in  the  eruptive  prominences  to  distances 
of  several  hundred  thousand  miles  (Fig.  157).  If  the  sun 
were  tidally  distorted,  the  eruptions  would  be  mostly  toward 
and  from  the  disturbing  sun ;  certainly  the  ejections  would 


Fig.  157.  —  Eruptive  prominence  at  three 
altitudes.  Photographed  by  Slocum  at 
the  Yerkes  Observatory. 


CH.  XII,  249]    EVOLUTION  OF   THE   SOLAR   SYSTEM     427 

reach  to  greater  distances  in  these  directions.  Besides  this, 
after  the  ejected  material  had  once  left  the  sun,  its  distance 
would  be  increased  still  further  by  the  attraction  of  S'. 
Consequently,  if  S'  were  not  moving  along  its  orbit,  the 
ejections  toward  and  from  it  would  be  to  more  remote  dis- 
tances than  they  would  be  in  any  other  direction.  In  fact, 
those  toward  S'  might  even  strike  it.  But  S'  would  be  mov- 
ipg  along  in  its  orbit,  and,  in  a  short  time,  it  would  have 
a  component  of  attraction  at  right  angles  to  the  original 
direction  of  motion  of  the  ejected  matter.  Consequently, 
by  the  time  S'  had  arrived  at  &'.  the  paths  of  the  ejected 
masses  would  be  curved  somewhat  hke  those  shown  in  Fig. 
1551  it  is  easy  to  see  that,  for  the  mass  ejected  toward "g". 
the  curvature  is  in  the  right  direction";  "a  discussion  based 
on  the  resolution  of  the  forces  involved  (Art.  153)  proves 
that,  for  the  mass  ejected  in  the  other  direction,  the  indicated 
curvature  is  also  correct.  Eventually  S'  would  move  on  in 
its  orbit  so  far  that  it  would  no  longer  have  sensible  attrac- 
tion for  the  ejected  masses,  and  they  would  be  left  revolving 
around  S  in  elhptical  orbits.  If  the  initial  speed  of  the 
ejected  material  were  very  great,  it  might  leave  S  never  to 
return. 

The  critical  question  is  whether  matter  woul5  be  ejected 
fay  enough  to  produce  the  large  orbits  required  by  the 
theory.  In  order  to  throw  light  on  this  question  the  follow- 
ing table  has  been  computed,  giving  the  surface  velocities 
necessary  to  cause  undisturbed  ejected  matter  to  recede 
various  distances  from  the  surface  of  the  sun. 

The  most  remarkable  thing  shown  in  the  table  is  that  after 
a  velocity  is  reached  sufficient  to  cause  the  ejected  matter 
to  recede  a  few  millions  of  miles,  a  small  change  in  the  initial 
speed  produces  radically  different  final  results.  Since  prom- 
inences now  ascend  to  a  height  of  half  a  million  of  miles 
without  the  disturbing  influence  of  a  visiting  sun,  it  is  seen 
that  the  numerical  requirements  of  the  hypothesis  are  not 
excessive.     Moreover,    numerous    actual    computations   of 


428     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xn,  z*» 


hypothetical  cases  have  shown  that,  on  the  recession  of  S', 
the  ejected  material  is  usually  left  revolving  around  S  in 
elliptical  orbits. 


Table  XIII 


Height  of 
Ascent 


100,000  mi 
200,000  mi, 
300,000  mi, 
400,000  mi, 
500,000  mi. 
1,000,000  mi, 
2,000,000  mi. 


Initial  Velocity 


72  mi 
121  mi 
157  mi 
184  mi 
206  mi 
268  mi 
316  mi 


per  sec. 
per  see. 
per  sec. 
per.  see. 
per  see. 
per  see. 
per  sec. 


Height  op  Ascent 


5,000,000  md. 

10,000,000  mi. 

20,000,000  mi. 

50,000,000  mi. 

100,000,000  mi. 

500,000,000  mi. 

Infinite 


Initial  Velocity 


353  mi. 
368  mi. 
376  mi. 
380  mi. 

382  mi. 

383  mi. 

384  mi. 


per  see. 
per  Fee. 
per  see. 
per  see. 
per  see. 
per  fee. 
per  see. 


As  one  star  passes  another  the  ejection  of  material  is  more 
or  less  continuous.  When  the  visiting  star  is  far  away,  the 
ejections  are  to  moderate  distances  and  the  matter  returns  to 

the  sun.  As  the  visit- 
ing star  approaches, 
the  ejected  materials 
recede  farther  and 
their  paths  become 
more  curved.  At  a 
certain  time  the  lat- 
eral  disturbance  of  S' 
becomes  so  great  that 
the'  ejected  material 
revolves  around  S  in- 
stead of  falhng  back 
upon  it.  Let  the 
orbits  for  this  case  be  those  marked!  and  1'  in  Fig.  158,  the 
former  being  toward  S',  and  the  latter  away  from  it.  At  a 
later  tiifie  the  ejections  will  be  to  greater  distances  and  the 
materials  will  have  greater  lateral  motions.  Suppose  they 
are  2  and  2',  and  so  on  for  still  later  ejections  until  S'  recedes 
from  iS. 


Fig.  158.  —  The  origiu  of  a  spiral  nebiila. 


CH.  XII,  249]   EVOLUTION  OF  THE   SOLAR  SYSTEM     429 

Now  consider  the  location  of  all  of  the  ejected  material 
at  a  given  time  after  S'  has  passed  its  nearest  point  to  S. 
If  it  has  been  sent  out  from  S  continuously,  it  will  He  along 
two  continuous  curves,  represented  by  the  full  lines  in  Fig. 


Fig.  159. — The  great  spiral  nebula  in  Canes  Venatici   (M.  51),  showing 
the  two  arms.      Photographed  by  Ritchey  at  the  Yerkes  Observatory. 

158.  These  are  the  arms  of  the  spiral  nebula  whose  indi- 
vidual particles  move  across  them  in  the  dotted  lines.  The 
diagram  shows  an  ideal  simple  case,  and  Fig.  159  an  actual 
photograph.  But  if  the  approach  of  S'  were  close,  or  if  there 
were  a  partial  collision,  and  if  the  ejected  material  should  go 
beyond  S',  a  very  complicated  structure  would  result.    The 


430     AN   INTRODUCTION   TO  ASTRONOMY    Ich.  xii,  ^» 

arms  of  the  spiral  might  be  very  irregular  (Fig.  160),  the 
particles  might  cross  them  at  a  great  variety  of  angles,  and 
some  of  them  might  continue  to  recede  indefinitely. 


Fig.  160.  —  The  great  spiral  nebula  in  Triangulum  (M.  33).     Photographed 
by  Ritchey  at  the  Yerkes  Observatory. 


Thus,  the  suggested  explanation  of  the  origin  of  the  spiral 
nebulae  rests  upon  the  existence  of  a  great  number  of  stars, 
their  rapid  and  somewhat  heterogeneous  motions  which 
imply  near  approaches  now  and  then,  their  eruptive  activities, 
and  the  disturbance  of  one  star  by  another  passing  near  it. 


CH.  XII,  250]   EVOLUTION  OP   THE    SOLAR   SYSTEM     431 

All  the  factors  involved  are  well  established  —  the  only  ques- 
tion is  that  of  their  quantitative  efficiency.  Here  some 
doubts  remain.  It  follows  from  the  number  of  stars,  the 
space  they  occupy,  and  their  motions  that,  if  they  were  mov- 
ing at  random,  an  individual  sun  would  pass  near  some  other 
one,  on  the  average,  only  once  in  many  thousands  of  millions 
of  years.  Perhaps  the  mutual  gravitation  of  the  stars  is 
important  out  on  the  borders  of  the  great  clusters  of  suns 
of  whieh  the  Milky  Way  is  composed,  where  it  may  reason- 
ably be  supposed  that  their  relative  velocities  are  small, 
and  it  may  be  that  in  these  regions  close  approaches  are  for 
this  reason  much  more  frequent.  But  in  any  case  the  demands 
of  time  are  very  formidable.  Besides  this,  many  of  the  spiral 
nebulae  are  of  such  enormous  dimensions  that  it  is  difficult 
to  suppose  they  have  been  produced  by  the  encounter  or 
near  approach  of  ordinary  suns.  It  may  be  stated,  however, 
that,  in  the  first  place,  there  is  no  positive  knowledge  what- 
ever respecting  the  masses  of  spiral  nebulae;  and  that,  in 
the  second  place,  near  approaches  are  not  confined  to  single 
stars,  but  may  involve  multiple  stars,  clusters,  and  systems 
of  stars.  The  observed  spirals  may  be  simply  the  larger 
examples  originating  from  several  or  many  suns. 

It  should  be  remembered  that,  whatever  doubts  may 
remain  respecting  the  validity  of  this  or  any  other  hypothesis, 
the  spiral  nebulae  certainly  exist  in  great  numbers,  and  they 
apparently  have,  on  an"  enormous  scale,  an  organization 
similar  to  that  which  we  have  inferred  must  have  been  the 
antecedent  of  the  solar  system.  And  it  may  be  stated  again 
that  the  planetesimal  hypothesis  rests  primarily  upon  the 
evidence  now  furnished  by  the  solar  system,  and  that  it  does 
not  stand  or  fall  with  any  theory  respecting  spiral  nebulae. 

250.  The  Origin  of  Planets.  —  According  to  the  planet- 
esimal hypothesis,  the  parent  nf  the  solar  svstem  con- 
sisted of  a  central  sun  surrounded  by  a  vast  swarm  of  plan- 
etesunals  which  moved  approximately  in  the  same  plane 
in  essentially  independent   elliptic   orbits.     Among  these 


432     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  250 

plapetesimals  there  were  nuclei,  or  local  centers  of  condensa- 
tion,  which,  in  their  revolutions,  swept  up  the  smaller  planet- 
esimals  and  grew  into  planets.  It  is  not  to  be  understood 
that  the  original  nuclei  were  solid  or  even  continuous  masses. 
It  is  much  more  probable  that  in  their  early  stages  they  were 
swarms  of  smaller  masses  having  about  the  same  motion 
with  respect  to  the  central  sun,  and  that,  undei*  their  mutual 
attractions  and  collisions,  they  gradually  condensed  into  con- 
tinuous  bodies.  Indeed,  the  condensation  may  have  been 
very  slow  and  may  have  been  dependent  to  an  irpportant 
extent  upon  the  impacts  of  other  planetesimals. 

It  seems  to  be  impossible  to  determine  the  probable  masses 
of  the  original  nuclei.  If  they  were  less  than  that  of  the 
moon  at  present,  they  could  not  have  retained  any  atmos- 
pheres under  their  gravitative  control.  But  as  the  nuclei 
grew,  their  surface  gravities  increased,  and  a  time  came 
when  those  which  have  become  the  larger  planets  possessed 
sufScient  gravitative  power  to  prevent  the  escape  of  atmos- 
pheric  particles.  The  acquisition  of  atmospheres  was  then 
inevitable  because,  in  the  first  place,  the  materials  grinding 
together  and  settling  under  the  weight  of  accumulating 
planetesimals  would  squeeze  out  the  Ughter  elements;  in 
the  second  place,  the  pulverizing  and  heating  effects  of  the 
impacts  of  meteors  would  liberate  gases ;  and,  in  the  third 
place,  the  growing  planets  in  their  courses  around  the  sun 
would  sweep  up  directly  great  numbers  of  atmospheric 
molecules.  The  extent  of  the  atmospheres  of  the  planets 
at  all  stages  of  their  growth  depended  primarily  on  their, 
surface  gravities . 

The  rate  at  which  the  nuclei  swept  up  the  planetesimals  must 
have  been  excessively  slow.  This  conclusion  follows  from  the 
fact  that  if  aU  the  matter  in  the  largest  planet  were  scat- 
tered around  the  sun  in  a  zone  reaching  halfway  to  the  ad- 
jacent planets,  the  resulting  planetesimals  would  be  very  far 
apart,  and  also  from  the  fact  that  the  orbits  of  only  a  fraction 
of  them  would  at  any  one  time  intersect  the  orbit  of  the 


CH.  xii,  251]   EVOLUTION  OF  THE   SOLAR  SYSTEM     433 

nucleus.  It  must  be  remembered  that  the  orbits  of  the  planet- 
esimals  were  continually  changed  by  their  mutual  attractions 
and  especially  by  the  attractions  of  the  nuclei.  Moreover, 
the  orbits  of  the  nuclei  were  continually  altered  by  collisions 
with  the  planetesimals  and  by  their  perturbations  of  one 
another.  Consequently,  if  the  orhitfi  of  tlip  nn^lpi  anrl  cer- 
tain  planetesimals  did  not  originally  intersect,  thev  might 
very  well  have  done  so  later.  But  it  does  not  follow  that 
they  have  all  been  swept  up  yet,  or,  indeed,  that  they  all 
ever  will  be  swept  up.  Possibly  some  of  the  meteors  which 
the  earth  now  encounters  are  the  stragghng  remains  of  the 
original  planetesimals. 

If  the  planetesimal  theory  is  correct,  the  earth  is  very  old 
and  the  sun  must  have  important  smn-(;;es  ftf  ^nRV-g-v  hpsidps 
its  contraction.  Most  of  the  geological  processes  did  not 
begin  until  it  became  large  enough  to  retain  water  and  an 
atmosphere.  These  same  conditions  were  necessary  for  even 
the  beginnings  of  the  development  of  life,  which  may  have 
had  a  continuous  existence  from  the  time  the  earth  was  half 
its  present  size. 

251.  The  Planes  of  the  Planetary  Orbits.  —  If  the  planet- 
esimal hypothesis  is  true,  it  must  explain  the  important 
features  of  the  solar  system.  The  most  striking  thing  about 
the  motions  of  the  planets  is  that  they  all  go  around  the  sun 
in  the  same  direction,  and  the  mutual  inclinations  of  the 
planes  of  their  orbits  are  small.  However,  some  deviations 
exist,  and  in  general  they  are  greatest  in  case  of  the  small 
masses  like  Mercury  and  the  planetoids. 

It  is  assumed  that  the  planetesimals  all  revolved  around 
the  sun  in  the  same  direction.  This  would  certainly  have 
been  true  if  they  originated  by  the  close  approach  of  two 
suns,  as  explained  in  Art.  249.  But  the  planes  of  their  orbits 
would  not  be  exactly  coincident.  The  plane  of  motion  of 
an  ejected  particle  would  depend  upon  its  direction  of 
ejection  and  the  forces  to  which  it  was  subject.  The  ejec- 
tions would  be  nearly  toward  or  directly  away  from  the  visit- 
2p 


434     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  251 

ing  sun,  but  slight  deviations  would  be  expected  because 
the  ejecting  body  might  be  rotating  in  any  direction,  and  the 
direction  of  ejection  would  depend  to  some  extent  upon  its 
rotation. 

Consider,  therefore,  a  central  body  surrounded  by  an 
enormous  swarm  of  planetesimals  which  move  in  intersecting 
elliptical  orbits,  some  close  to  the  sun  and  others  far  away. 
The  system  of  planetoids  now  in  the  solar  system  gives 
a  fair  picture  of  the  hypothetical  situation,  especially  if,  as 
seems  very  probable,  there  are  countless  numbers  of  small 
ones  which  are  invisible  from  the  earth.  Suppose,  also,  that 
there  exist  a  number  of  nuclei  revolving  at  various  distances. 
They  gradually  sweep  up  the  smaller  masses,  and  the  problem 
is  to  determine  what  happens  to  the  planes  of  their  orbits. 

Consider  a  nucleus  and  all  the  planetesimals  which  it  will 
later  sweep  up.  AH  together  they  have  what  may  be  called 
in  a  rough  way  an  average  plane  of  revolution.  This  is  a 
perfectly  definite  dynamical  quantity  which  Laplace  treated 
and  which  he  called  the  "  invariable  plane." 

When  all  the  masses  have  united,  the  resulting  body  will 
inevitably  revolve  in  this  plane.  If  the  nucleus  originally 
moved  in  some  other-plane,  the  plane  of  its  orbit  would  con- 
tinually change  as  its  mass  increased.  The  same  would  be 
true  for  every  other  nucleus.  There  would  be  also  an  aver- 
age plane  for  the  whole  system.  Those  nuclei  which  moved 
in  regions  that  were  richest  in  planetesimals,  and  that  grew 
the  most,  would,  in  general,  have  final  orbits  most  nearly 
coincident  with  this  average  plane.  It  is  clear  that  so  far 
as  the  planes  of  the  orbits  of  the  planets  are-  concerned 
(see  Table  IV),  the  consequences  of  the  planetesimal  theory 
are  in  perfect  harmony  with  the  facts  estabhshed  by 
observation. 

252.  The  Eccentricities  of  the  Planetary  Orbits.  —  The 
orbits  of  the  original  planetesimals  probably  had  a  consider- 
able range  of  eccentricities.  This  view  is  supported  by  the 
fact  that  the  eccentricities  of  the  orbits  of  the  planetoids  vary 


v-a.  ^ii,  .<iui,j    juvuiju  iiUiNi    UF   TriE    SOLAR  SYSTEM     435^ 

from  nearly  zero  to  about  0.5.  It  is  also  supported  by  the 
computations  of  orbits  of  particles  which  were  assumed  to 
be  ejected  from  one  sun  when  another  was  passing  it.  The 
problem  is  to  find  whether  nearly  circular  planetary  orbits 
would  be  evolved  from  such  a  system  of  planetesimals. 

When  a  nucleus  sweeps  up  a  planetesimal,  the  impact  on 
the  larger  body  may  be  in  any  direction.  If  the  nucleus 
overtakes  the  planetesimals  so  that  they  act  like  a  resisting 
mediiun,  the  eccentricity  of  its  orbit  is  in  general  diminished, 
as  was  proved  by  Euler  more  than  150  years  ago.  But  many 
other  kinds  of  encounters  can  occur  between  bodies  all 
moving  in  the  same  direction  around  the  sun.  Colhsions  will 
obviously  be  most  numerous  between  bodies  whose  orbits 
are  approximately  of  the  same  dimensions ;  if  the  orbits  of 
two  bodies  differ  greatlv  in  size,  collision  betwe^p  them  is 
impossible  unless  the  orbits  are  verv  elongated.  It  is  a  re- 
markable  general  proposition  that  if  two  bodies  are  moving  in 
orbits  of  the  same  size  and  shape,  but  differently  placed,  and 
if  they  colUde  in  any  way,  the  eccentricity  of  the  orbit  of 
the  combined  mass  will  be  smaller  than  the  common  eccen- 
tricity of  the  orbits  of  the  separate  parts.' 

1  To  prove  this,  suppose  a  nucleus  M  and  a  planetesimal  m  are  moving  in 
orbits  whose  major  semi-axis  and  eccentricity  are  ao  and  eo.  Let  their 
velocities  at  the  instant  preceding  collision  be  Va  and  so,  and  their  combined 
velocity  after  collision  be  V.  The  kinetic  energy  of  the  two  bodies  at  the 
instant  preceding  collision  is  i(MVo^  +  mvo^).  Their  kinetic  energy  after 
their  union  is  i{M  +  m)V^.  The  latter  will  be  smaller  than  the  former 
because  some  energy  will  have  been  transformed  into  heat  by  the  impact  of 
the  two  parts.     Therefore  iWFo^  +  mvo^  >  (M  +  m)  V. 

It  is  shown  in  celestial  mechanics  in  the  problem  of  two  bodies  that  in 

2      1 
elliptic    orbits    F^  =  —  -.     Hence,  the    inequality    becomes 
r     a 

M(2-l)+»(2-L)>(M+™)(2-i), 
\r     oo'  ^r      oo'  ^  r      a' 

where  a  is  the  major  semi-axis  of  the  combined  mass.  It  follows  from  this 
inequality  that  M+jn^M_±m,^   whence  a<m.     That  is,  under  the  cir- 

cumstances  of  the  problem  a  collision  always  reduces  the  major  semi-axis  of 
the  orbit. 

Another  principle  established  in  celestial  mechanics  is  that  the  moment 


436     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  252 

Of  course,  if  two  orbits  were  of  exactly  the  same  size,  the 
periods  of  the  bodies  would  be  the  same  and  collisions  would 
result  either  at  the  first  revolution  or  only  after  their  mutual 
attractions  had  modified  their  motions.  But  if  they  were 
of  nearly  the  same  size,  the  conditions  for  colhsions  would  be 
favorable,  and  in  nearly  all  cases  the  eccentricity  would  be 
reduced. 

It  follows  from  this  discussion  that,  in  general,  colUsions 
between  planetesimals  cause  the  eccentricities  of  their  orljiits 
to  decrease!  Consequently,  the  more  a  nucleus  growsTy 
sweeping  up  planetesimals,  the  more  nearly  circular,  in  general, 
its  orbit  will  be.  If  a  nucleus  revolves  in  a  region  rich  iu 
planetesimals,  the  result  is  likely  to  be  a  large  planet  whose 
orbit  has  small  eccentricity.  These  conclusions  agree  pre- 
cisely with  what  is  found  in  the  solar  system,  for  the  orbits  of 
all  the  large  planets  are  nearly  circular,  while  tne  oroits  oi 
some  of  the  smaller  planets  and  many  of  the  planetoids  are 
considerably  eccentric.  ^^ 

253.  The  Rotation  of  the  Sun.  —  If  the  central  body  in 
the  planetesimal  system  rotates  in  the  direction  of  the  motion 
of  the  outlying  parts,  the  final  result  will  be  a  sun  rotating 
in  the  direction  of  revolution  of  its  planets.  But  if  the 
planetesimal  organization  is  the  result  of  the  close  approach 
of  two  suns,  the  central  mass  might  originally  have  been 
rotating  in  any  direction.  In  this  case  the  final  outcome 
is  not  quite  so  obvious. 

The  only  planetesimals  which  could  sensibly  affect  the 
rotation  of  the  central  mass  are  those  which  fall  back  upon 
it.     If  the  planetesimals  originated  by  the  close  approach 

of  momentum  is  constant  whether  there  are  collisions  or  not.  The  orbital 
moment  of  momentum  of  a  mass  m  is  mvo(l  — e^),  where  e  is  the  eccen- 
tricity. The  condition  that  the  moment  of  momentum  before  collision 
shall  equal  that  after  collision  is,  therefore, 


MVaail  -e«')  +  mVao(l  -co^)   =  (M  +  m)  VaCl-e^),   or 

Vao(l  -eo2)   =  VaCl  -e^). 
Since  Oo  >  a,  it  follows  that  V(l  —  eo^X  Vl  —e'',  and  therefore  that  e  <  eo. 


CH.  XII,  }iO^   JiiVULiUTlON  OF   THE    SOLAR  SYSTEM     437 

of  two  suns,  there  would  certainly  be  many  which  would 
return  to  the  central  mass.  They  would  not  fall  straight  in 
towards  its  center,  but  would  have  a  small  forward  motion 
similar  in  character  to  that  of  the  remainder  of  the  planet- 
esimals.  The  result  of  the  colhsion  would  be  that  the  sun 
would  acquire  their  moment  of  momentum.  It  does  not 
seem  unreasonable  that  the  mass  of  the  central  sun  might 
grow  in  this  way  by  as  much  as  10  per  cent.  Since  the  planet- 
esimals  would  have  enormously  more  moment  of  momen- 
tum than  equal  masses  in  the  central  body,  they  would 
substantially  determine  its  direction  of  rotation.  In  fact, 
if  they  were  moving  in  orbits  whose  eccentricity  was  0.9 
and  if  they  just  grazed  the  sun  at  their  perihelion,  the  mass 
necessary  to  account  for  the  present  rotation  of  the  sun,  if 
it  had  no  rotation  originally,  would  be  one  fifth  of  one  per 
cent  of  the  sun's  mass. 

Another  interesting  result  remains  to  be  mentioned.  The 
planetesimals  would  strike  the  equatorial  region  of  the  sun 
in  greatest  abimdance  and  would  give  it  the  most  rapid 
motion.  Unless  the  inequalities  in  motion  were  worn  down 
by  friction  the  equatorial  zone  would  be  rotating  fastest,  as 
is  the  case  with  our  own  sun. 

254.  The  Rotations  of  the  Planets.  —  The  earth.  Mars, 
Jupiter,  and  Saturn  rotate  in  the  direction  in  which  the 
planets  revolve ;  the  surfaces  of  the  other  planets  have  not 
been  observed  well  enough  to  enable  astronomers  to  deter- 
mine how  they  rotate.  It  has  been  generally  supposed  that 
the  equators  of  Uranus  and  Neptune  coincide  with  the 
planes  of  the  orbits  of  their  satellites,  but  the  evidence  in 
support  of  the  supposition  is  as  yet  inconclusive. 

The  earlier  theories  regarding  the  origin  of  the  planets  all 
fail  to  explain  their  forward  rotations. 

ChamberKn  has  shown  that  if  a  planet  develops  from 
a  planetesimal  system  it  will  m  general  rotate  m  the  direc- 
tion  of  its  revolution.  Consider  a  nucleus  N,  Fig.  161, 
^^ch,  m  its  early  stages,  will  probably  be  simply  an  immense 


438     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  254 

swarm  of  planetesimals.  For  simplicity,  suppose  its  orbit 
is  a  circle  C  around  the  sun  as  a  center  (if  this  assumption 
were  not  made,  the  discussion  would  not  be  essentially  modi- 
fied). The  planetesimals  which  can  encounter  iV  are  divided 
into  three  classes:  (a)  those  whose  aphelion  pomts  are 
inside  the  circle  C ;  (b)  those  whose  perihelion  points  are 
inside  C  and  whose  aphehon  points  are  outside  of'C;   and 

(c)  those  whose  perihe- 
lion points  are  outside  of 
C.    They  are  designated 
"\^      by  (a),  (b),  and  (c)  re- 
"\     spectively  in  Pig.  161. 
\        Consider  collisions  of 
;   the     planetesimals      of 
;'   class  (a)  with  the    nu- 
/    cleus    A''.      A     collision 
can  occur  only  when  a 
planetesimal  is  near  its 
aphelion  point.     At  and 
near     this     point     the 

Fig.    161.  —  Development  of   the    forward    planetesimal    is   mOving 
rotation  of  a  planet  nucleus  by  the  accre-      j  ^-^^^      ^^^      ^^_ 

tion  of  planetesimals. 

cleus.i 

Hence  the  nucleus  will  overtake  the  planetesimal,  and  the 
collision  will  be  a  blow  backward  on  the  inner  side  of  the 
nucleus.  That  is,  planetesimals  of  class  (a)  tend  to  give  the 
nucleus  a  forward  rotation . 

Planetesimals  of  class  (6)  can  strike  the  nucleus  so  as  to 
tend  to  give  it  a  rotation  in  either  direction,  or  so  as  not  to 
have  any  effect  on  its  rotation.  If  they  are  not  distributed 
in  some  special  way,  the  collective  result  of  the  collision  of 
many  of  them  will  be  very  small. 

>  Let  V  and  v  represent  the  velocity  of  the  nucleus  and  planetesimal 
respectively,  and  A  and  a  the   semi-axes  of   their  orbits.     It  is  shown  in 

9  1  2  1 
celestial  mechanics  that  V^  =- ,  and  v^  = .     Since  a<  A  and  r  is 

r      A  r      a 

the  same  in  the  two  equations,  it  follows  that  V  >  v^. 


CH.  XII,  264]   EVOLUTION  OF   THE   SOLAR  SYSTEM     439 

Planetesimals  of  class  (c)  move  faster  than  the  nucleus 
at  the  time  of  collision.  Therefore  they  overtake  the  nu- 
cleus and  tend  to  give  it  a  forward  rnta.tinn . 

It  follows  from  this  discussion  that  two  of  the  three  classes 
of  planetesimals  tend  to  give  the  nucleus  a  forward  rotation. 
The  effects  are  most  important  at  the  equator  of  the  planet, 
for  there  they  strike  farthest  from  its  axis.  Hence,  the  im- 
pacts of  planetesimals  on  the  whole  tend  to  make  the  equa- 
tors of  fluid  planets  rotate  faster  than  the  higher  latitudes,  as 
is  the  dase  with  Jupiter  and  Saturn.  The  precise  final  result 
depends  upon  the  initial  rotation  of  the  nucleus  and  upon  the 
distribution  of  the  planetesimals  among  the  three  classes. 

Obviously  the  relative  numbers  of  planetesimals  in  classes 
(a)  and  (c)  would  in  general  be  small.  In  order  to  get  some 
idea  of  the  numbers  required  to  account  for  the  observed 
rotations,  a  numerical  example  has  been  treated.  It  was 
assumed  that  the  original  earth  nucleus  had  no  rotation 
and  that  the  planetesimals  of  class  (&)  gave  it  none.  It 
was  assumed  that  all  the  planetesimals  of  classes  (a)  and 
(c)  moved  in  orbits  having  the  eccentricity  0.2  and  that  they 
struck  the  nucleus  4000  miles  from  the  center.  Then,  in 
order  to  account  for  the  present  rotation  of  the  earth,  it  was 
found  that  their  total  mass  must  have  been  about  5.7  per 
cent  of  that  of  the  whole  earth.  Whether  or  not  these 
results  are  reasonable  cannot  be  determined  without  further 
quantitative  investigations.  But  it  must  be  insisted  that 
the  results  are  qualitatively  correct,  and  that  not  even  this 
much  can  be  said  for  any  earlier  hypothesis  regarding  the 
origin  of  the  planets. 

In  the  preceding  discussion  the  effects  of  the  rotations  of 
the  original  nuclei,  or  swarms  of  planetesimals  out  of  which 
the  nuclei  condensed,  have  been  ignored.  As  a  matter  of 
fact,  they  were  probably  in  rotation  around  axes  essentially 
perpendicular  to  the  plane  of  the  system.  There  seems  to 
be  no  conclusive  reason  why  the  original  rotations  should 
have  been  in  one  direction  rather  than  in  the  other,    The 


I 
440     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  254 

observed  rotations  of  the  planets  seem  to  indicate  that, 
for  some  reason  at  prfsi^nt.  iinlfnown^  fhp  original  niif^lpi 
rotated  in  the  forward  direction . 

255.  The  Origin  of  Satellites.  —  According  to  the  planet- 
esimal  theory,  the  satellites  developed  either  from  small 
secondary  nuclei  which  were  associated  with  the  larger 
planetary  nuclei  from  the  beginning,  or  froni  neighboring 
secondary  nuclei  which  became  entangled  at  a  later  time  in 
the  outlying  parts  of  the  swarms  of  planetesimals  constitut- 
ing the  nuclei.  If  the  satellites  originated  in  the  former  way, 
their  directions  of  revolution  would  be  the  same  as  those  of 
rotation  of  their  respective  primaries ;  if  in  the  latter  way, 
they  might  revolve  originally  in  any  directions  aroimd  their 
primaries. 

With  the  exception  of  the  eighth  and  ninth  satellites  of 
Jupiter  and  the  ninth  satellite  of  Saturn  (and  possibly  the 
satellites  of  Uranus  and  Neptune),  all  the  known  satellites 
revolve  in  the  directions  in  which  their  primaries  rotate. 
This  seems  to  indicate  that  at  least  most  of  the  satellites 
originated  from  secondary  nuclei  which  were  associated  with 
their  respective  prunarv  nuclei  fr9m  the  heinnninp-  and  par- 
took  of  their  common  raotion  of  rotation.  The  satellite 
nuclei,  hke  the  planetary  nuclei,  swept  up  the  planetesimals 
and  grew  in  mass.  The  craters  on  the  moon  may  have 
been  produced  by  the  impact  of  planetesimals. 

With  the  growth  in  mass  of  a  planet  its  attraction  for  its 
satelUtes  increases  and  this  results  in  a  reduction  in  the 
dimensions  of  their  orbits.  Suppose  the  most  remote  direct 
satellites  were  originally  revolving  at  the  greatest  distances 
at  which  their  primaries  could  hold  them  in  gravitative  con- 
trol, and  that  their  orbits  have  been  reduced  to  their  present 
dimensions  by  the  growth  of  the  planets.  The  amount  of 
reduction  in  the  size  of  the  orbit  of  a  satellite  depends  upon 
the  amoimt  of  growth  of  the  planet  around  which  it  revolves, 
and  furnishes  the  basis  for  computing  the  increase  in  the 
mass  of  the  planet. 


CH.  XII,  256]   EVOLUTION  OF  THE   SOLAR  SYSTEM     441 

The  three  retrograde  satellites  revolve  at  great  distances 
from  their  respective  primaries  in  orbits  which  are  rather 
eccentric  and  considerably  incUned  to  their  respective  sys- 
tems. Their  origin  is  evidently  different  from  that  of  the 
direct_satelUtes.  They  may  have  been  neighboring  planet- 
esimals  which  became  entangled  in  the  remote  parts  of  the 
plapetary  swarm.  The  question  arises  whv  thev  revolve 
in/ the  retrograde  direction.  The  answer  probably  depends 
upon  the  fact  that,  at  a  given  distance,  a  retrograde  sateUite 
is  much  more  stable,  so  far  as  the  disturbance  of  the  sun  is 
concerned,  than  a  direct  one.  Consequently,  a  retrograde 
satellite  would  not  be  driven  by  collisions  away  from  the  con- 
trol of  its  planet  so  easily  as  a  direct  one.  Also,  the  effects  of 
collisions  with  planetesimals  and  satelUtesimals  (planetesimals 
revolving  around  planetary  nuclei)  must  be  considered. 

256.  The  Rings  of  Saturn.  —  The  rings  of  Saturn  are 
swarms  of  particles  revolving  in  the  plane  of  the  planet's 
equator.  According  to  the  planetesimal  theorv,  thev  are 
the  remains  of  outlying  masses  in  the  original  nucleus  which 
were  moving  so  fast  that  they  did  not  fall  toward  the  center. 
Of  course,  they  were  subject  to  encounters  with  in-falUng 
planetesimals.  These  colUsions  transformed  some  of  their 
energy  of  motion  into  heat  and  some  of  them  fell  toward, 
or  perhaps  on  to,  the  growing  planetary  nucleus.  It  may 
be  that  only  a  small  part  of  the  original  ring  material  now 
remains.  But  when  they  fell,  they  retained  at  least  a  portion 
of  their  motion  of  revolution,  and  the  result  was  that  they 
struck  the  planet  obliquely  in  the  direction  in  which  it 
rotated.  This  increased  its  rotation,  especially  in  the  plane 
of  its  equator. 

There  may  be  and  probably  are  collisions  even  now 
among  the  particles  which  constitute  the  rings  of  Saturn. 
If  there  are  colUsions,  the  energy  of  motion  is  being  trans- 
formed into  heat,  and  this  comes  from  the  energy  of  the 
orbital  motions,  with  the  result  that  the  dimensions  of  the 
rings  are  being  decreased.    They  may  ultimately  disappear 


442     AN   INTRODUCTION   TO   ASTRONOMY    [ch.  xii,  jsod 

for  this  reason,  and  it  is  not  impossible  that  other  planets 
also  once  had  ring  systems. 

257.  The  Planetoids.  —  The  planetoids  occupy  a  zone  in 
which  there  was  no  predominating  nucleus.  They  probably 
have  not  grown  so  much  relatively  as  the  planets  by  the 
accretion  of  planetesimals.  Hence  the  ranges  in  the  eccen- 
tricities and  inchnations  of  their  orbits  give  a  better  idea 
of  the  character  of  the  orbits  of  the  original  planetesimals. 

Besides  the  known  planetoids,  there  are  probably  thou- 
sands of  others  which  are  so  small  that  they  have  not  been 
seen.  There  may  be  others  also  between  the  orbits  of 
Jupiter  and  Saturn  and  beyond  the  orbit  of  Saturn.  At 
those  vast  distances  none  but  large  bodies  would  be  visible, 
both  because  they  would  not  be  strongly  illuminated  by 
the  sun  and  also  because  they  would  always  be  very  remote 
from  the  earth.  The  planetoid  Eros  has  escaped  collision 
with  Mars  only  because  of  the  inclination  of  its  orbit.  It 
is  not  unreasonable  to  suppose  that  there. are  many  other 
planetesimals  between  the  orbits  of  the  earth  and  Mars 
which  are  too  small  to  be  visible. 

258.  The  Zodiacal  Light.  —  It  is  universally  agreed  that 
the  zodiacal  light  is  due  to  a  great  swarm  of  small  bodies, 
or  particles,  revolving  around  the  sun  near  the  plane  of  the 
earth's  orbit.  These  small  bodies  are  in  reality  planetesi- 
mals which  have  not  been  swept  up  by  the  planets  either 
because  of  the  high  inclination  of  their  orbits,  or  more 
probably  because  their  orbits  are  so  nearly  circular  that  they 
do  not  cross  the  orbits  of  any  of  the  planets. 

259.  The  Comets.  —  Recent  investigations  have  shown 
that  it  is  very  probable  that  comets  are  permanent  members 
of  the  solar  system.  As  they  have  no  intimate  relationship 
to  the  planets,  the  question  of  their  origin  presents  new 
problems  and  difficulties. 

According  to  the  planetesimal  theory,   the  comets  are^ 
possibly  only  the  outlying  and  tenuous  fragments  of  the  orig- 
inal nebula  which  did  not  partake  of  the  general  rotation 


CH.  xii,  .iou]    JiiVULiUTlON   OF   THE    SOLAR  SYSTEM     443 

of  the  system.  If  the  planetesimal  system  was  produced  by 
the  near,  approach  of  two  suns,  t.hpy  may  have  had  their 
origin,  as  ChamberUn  has  suggested,  in  the  dispersion  and 
scattering  of  eariier  planetesimals  which  revolved  in  different 
planes ;  or  there  may  have  been  explosions  of  lighter  ya.sps 
in  various  directions,  which,  under  the  disturbing  action  of 
the  visitmg  sun,  did  not  fall  back  upon  our  own ;  or  the 
comets  may  be  matter  which  was  ejected  from  the  visiting 
sun.  Th£  differences  in  the  lengths  of  their  orbits  and  in  the 
positions  of  the  planes  of  their  orbits  may  originally  have 
been  much  less  than  at  present,  for  the  planets  may  have  dis- 
turbed their  motions  to  almost  any  extent.  The  planets 
may  have  captured  some  comets  and  greatly  enlarged  the  or- 
bits of  an  equal  number  of  others,  and  they  may  have  entirely 
changed  the  positions  of  the  planes  of  the  cometary  orbits. 

260.  The  Future  of  the  Solar  System.  —  The  theory  has 
been  developed  that  the  planets  have  grown  up  from  nuclei 
by  the  accretion  of  scattered  planetesimals.  They  acquired 
and  retained  atmospheres  when  4heir  masses  became  great 
enough  te  prevent  the  escape  of  gases,  mnlennle  by  mnlpnnlp. 
Their  masses  are  still  increasing,  but  the  process  of  growth 
seems  to  be  essentially  finished.  Those  planets  which  are 
dense  and  solid  Uke  the  earth  will  retain  all  their  essential 
characteristics  as  long  as  the  sun  continues  to  furnish  radiant 
energy  at  its  present  rate.  The  large  rare  planets  will  lose 
heat  from  their  interiors  and  may  contract  appreciably. 
The  reason  that  loss  of  heat  may  be  important  for  them  and 
not  for  the  sohd  planets  is  that  it  can  be  carried  to  the  sur- 
face rapidly  by  convection  in  a  gaseous  or  liquid  body,  while 
in  a  sohd  body  it  is  transferred  from  the  interior  only  by  the 
excessively  slow  process  of  conduction. 

The  duration  of  the  sun  is  a  very  important  factor  in  the 
future  of  the  planets.  There  is  no  known  source  of  energy 
which  could  supply  its  present  rate  of  radiation  many  tens 
of  millions  of  years.  Yet  it  is  not  safe  to  conclude  that  the 
sun  will  cool  off  in  a  few  miUions  of  years  because  the  earth 


444     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  ^ou 

gives  indisputable  evidences  (Art.  219)  that  the  svin  has 
radiated  more  energy  than  could  have  been  supplied  by  any 
known  source.  The  existence  of  hundreds  of  millions  of 
stars  blazing  in  full  glory  also  suggests  strongly  that  the 
lifetime  of  a  sun  is  very  long,  for  it  is  not  reasonable  to  sup- 
pose that,  if  they  endured  only  a  comparatively  short  time, 
so  many  of  them  would  now  have  such  great  brilUancy.  In 
view  of  these  uncertainties  it  is  not  safe  to  set  any  definite 
Umit  on  the  future  duration  of  the  sun,  however  probable 
its  final  extinction  may  be. 

If  the  sun  cools  off  before  something  destroys  the  planets, 
they  will  revolve  aroimd  it  cold,  lifeless,  and  invisible,  while 
it  pursues  its  journey  through  the  trackless  infinity  of  space. 
If  the  radiation  of  the  sun  does  not  sensibly  diminish,  the 
earth,  and  possibly  some  of  the  other  planets,  will  continue 
to  be  suited  for  the  abode  of  life  until  they  are  in  some 
way  destroyed.  Whether  or  not  the  sun  becomes  cold,  the 
planets  will  be  broken  into  fragments  when  our  sun  passes 
sufficientlv  near  another  star.  Their  remains  mav  then  be 
dispersed  among  the  revolving  masses  of  a  new  planetesimal 
system,  to  become  in  time  parts  of  new  planets.  Indeed, 
the  meteorites  which  now  strike  the  earth  often  give  evidence 
of  having  once  been  in  the  interior  of  masses  of  planetary 
dimensions,  and  Chamberlin  has  suggested  that  they  may 
be  the  remains  of  a  family  of  planets  antedating  our  own. 
To  such  dizzy  heights  are  we  led  in  sober  scientific  pursuits  ! 

The  question  of  the  purpose  of  all  the  wonderful  things  in 
the  universe  is  one  which  ever  arises  in  the  human  mind. 
With  subUme  egotism  men  have  usually  answered  that  every- 
thing was  created  for  their  pleasure  and  benefit.  The  sun 
was  made  to  give  them  fight  by  day,  and  the  moon  and  the 
myriads  of  stars  to  illuminate  their  way  by  night.  The 
numberless  plants  and  animals  of  forest  and  prairie  and 
sea  were  supposed  to  exist  primarily  for  the  profit  of  the 
human  race.  But  with  the  increase  of  knowledge  this  con- 
clusion is  seen  to  be  absurd.    How  infinitesimal .  a  part  of 


CH.  XII,  ijouj   JUVOLUTION  OF  THE   SOLAR  SYSTEM     445 

the  solar  system  and  its  energy  man  can  use,  to  say  nothing 
of  that  in  the  hundreds  of  millions  of  other  systems  which 
are  found  in  the  sky  ! 

How  many  bilUons  of  creatures  were  born,  lived,  and  died 
before  man  appeared  !  The  precise  time  of  the  beginning  of 
life  on  the  earth  and  the  manner  of  its  origin  are  lost  in  the 
distant  past.  In  the  oldest  rocks  laid  down  as  sediments 
tens  of  millions  of  years  ago  in  the  Archeozoic  era  there  are 
indications  of  the  existence  of  low  forms  of  Ufe  on  the  earth. 
In  the  Cambrian  period  trilobites  and  other  lowly  creatures 
lived  in  great  abimdance.  In  the  Ordovician  period  the 
types  of  low  forms  greatly  increased  and  the  vertebrates 
began  to  appear ;  in  the  Silurian,  they  were  firmly  established ; 
in  the  Devonian,  they  were  still  further  developed.  And 
after  many  other  geological  periods  had  passed,  the  higher 
forms  of  life,  including  man,  appeared.  Now  man  finds 
himself  a  part  of  this  great  life  stream,  not  something  funda- 
mentally different  from  the  rest  and  that  for  which  it  exists. 
If  the  earth  shall  last  some  millions  or  tens  of  millions  of 
years  in  the  future,  as  seems  Ukely,  the  physical  and  mental 
changes  which  the  human  race  will  imdergo  may  be  as  great 
as  those  through  which  the  animal  kingdom  has  passed  during 
the  long  periods  of  geological  time.  This  is  especially  prob- 
able if  men  learn  how  to  direct  the  processes  of  .their  own 
evolution.  But  if  they  do  not,  the  human  race  may  become 
extinct  just  as  many  other  species  of  animals  have  become 
extinct.  However  this  may  be,  it  seems  certain  that  its  end 
will  come,  for  eventually  the  fight  of  the  sun  will  go  out,  or 
the  earth  and  the  other  planets  will  be  wrecked  by  a  passing 
star,  and  the  question  of  the  purpose  of  it  all,  if  indeed 
there  is  any  purpose  in  it,  still  remains  unanswered. 

XX.     QUESTIONS 

1.  Are  the  particles  which  produce  the  zodiacal  light  an  example 
of  the  planetesimal  organization? 

2.  In  the  case  of  one  star  passing  by  another,  why  would  their 
ejections  of  material  be  largely  toward  or  from  each  other? 


446     AN   INTRODUCTION   TO  AbTituiN  uivj.  i    lv^^.  ^.., 

3.  Show  by  a  resolution  of  the  forces  that  the  material  ejected 
both  toward  and  from  S'  wiU  describe  curves  around  S  in  the  same 
direction. 

4.  WiU  the  orbit  of  S'  be  changed  if  it  changes  the  moment  of 
momentum  of  the  system  S  ?  What  wiU  be  the  result  in  the  very 
special  case  where  the  orbit  of  S'  relatively  to  S  is  originally  a 
parabola  ? 

5.  In  view  of  Table  XIII,  what  fraction  of  the  material  ejected 
from  S  would  reasonably  be  expected  (a)  to  fall  back  on  S,.  (6)  to 
revolve  around  it  in  the  planetesimal  state,  (c)  to  escape  from  its 
gravitative  control  ?  On  the  basis  of  these  figures,  find  what  frac- 
tion of  /S  would  need  to  be  ejected  altogether  in  order  to  provide 
material  for  the  planets. 

6.  Would  the  eccentricities  of  the  orbits  of  the  material  which  feU 
back  upon  S  have  been  large  or  small  ?  Would  most  of  the  collisions 
have  been  grazing,  as  was  assumed  in  the  discussion  in  Art.  253  ? 

7.  In  view  of  the  kinetic  theory  of  gases,  would  a  gaseous  nucleus 
as  massive  as  the  moon  concentrate  or  dissipate  ?  Would  a  nucleus 
of  the  materials  found  in  the  sun  remain  gaseous  on  cooling  ? 

IV.  Historical  Cosmogonies 

261.  The  Hypothesis  of  Kant.  —  The  work  of  Thomas 
Wright,  which  preceded  that  of  Kant  by  five  years,  was 
concerned  chiefly  with  the  evolution  of  the  whole  sidereal 
universe.  Wright  supposed  the  Milky  Way  is  made  up  of 
a  great  number  of  mutually  attracting  systems  which  are 
spread  out  in  a  great  double  ring  rotating  about  an  axis 
perpendicular  to  its  plane.  Kant  was  the  first  one  to  imder- 
take  the  development  of  a  detailed  theory  of  the  evolution  of 
the  solar  system  on  the  basis  of  the  law  of  gravitation. 

Kant's  interest  in  cosmogony  was  aroused  by  the  book 
of  Wright,  which  fell  into  his  hands  in  1751.  He  at  once 
turned  his  keen  and  penetrating  mind  to  the  question  of  the 
origin  of  the  planets,  and  wrote  a  briUiant  work  on  the  sub- 
ject. On  almost  every  page  he  gave  proof  of  the  intellectual 
power  which  later  made  him  the  foremost  philosopher  of  his 
time,  yet  his  theories  were  not  without  serious  imperfections. 

Kant  postulated  that  the  parent  of  the  solar  system  was 
a  vast  aggregation  of  simple  elements,  without  motion  and 


CH.  XII,  261]   EVOLUTION  OF  THE   SOLAR  SYSTEM     447 

subject  only  to  gravitational  and  chemical  forces  and  the 
repulsion  of  molecules  in  a  gaseous  state.  Nothing  could 
have  been  simpler  for  a  start.  The  problem  was  to  show  how 
such  a  system  could  develop  into  a  central  sun  and  a  family 
of  widely  separated  planets. 

Kant  reasoned  that  motions  among  the  molecules  would 
be  set  up  by  their  chemical  affinities  and  mutual  attractions. 
He  stated  that  the  large  molecules  would  draw  to  themselves 
the  smaller  ones  in  their  immediate  neighborhood,  and  that 
with  growth  their  power  of  growing  would  continually 
increase.  He  believed  that  not  only  would  aggregations  of 
molecules  be  formed,  but  that  these  masses  would  acquire 
motions  both  because  of  the  attraction  of  the  system  as  a 
whole  and  also  because  of  their  mutual  attractions.  Kant 
called  attention  to  the  fact  that  attraction  would  be  opposed 
by  gaseous  expansion,  and  he  supposed  that  these  repulsive 
forces  in  some  obscure  way  would  generate  lateral  motipns 
in  the  small  nuclei.  At  first  the  nuclei  would  be  moving 
in  every  possible  direction,  but  he  assumed  that  successive 
collisions  woujd  eliminate  all  except  a  few  moving  in  the 
same  direction  in  nearly  circular  orbits. 

The  beauty  and  generality  of  Kant's  theory  are  enticing, 
but  it  involves  some  obvious  and  fatal  difficulties.  In  the 
first  place,  the  attractive  and  repulsive  forces  would  not 
be  competent  to  set  up  a  general  revolution  of  a  system 
which  was  originally  at  rest.  His  conclusion  in  this  matter 
squarely  violates  the  principle  that  the  moment  of  momentum 
of  an  isolated  system  remains  constant. 

Notwithstanding  clear  statements  by  Kant,  some  writers 
have  modified  his  theory  by  supposing  that  there  was  hetero- 
geneous motion  of  the  original  chaos  with  a  predominance  in 
the  direction  in  which  the  planets  now  revolve.  But  with 
this  concession  to  the  theory,  which  makes  it  dynamically  a 
different  theory,  difficulties  still  remain.  It  is  not  at  all 
clear  that  in  a  system  of  such  enormous  extent  the  orbits 
of  all  bodies  except  those  having  motion  in  the  dominant 


448     AN   INTRODUCTION  TO  ASTRONOMY    [ch.  xii,  261 

direction  would  be  destroyed  by  collisions.  There  is,  Indeed, 
no  apparent  reason  why,  if  this  were  the  true  history  of  the 
origin  of  the  planets,  some  planets  should  not  now  be  found 
revolving  at  right  angles  to  the  general  plane  of  the  system, 
or  even  in  the  retrograde  direction.  This  is  not  impossible, 
as  is  proved  by  the  motions  of  the  comets.  Thus  it  is  seen  that 
if  Kant's  hypothesis  is  taken  strictly  as  he  gave  it,  the  condi- 
tion that  the  moment  of  momentum  of  the  system  should 
have  its  present  value  is  violated,  and  that  if  the  postulates 
are  changed  so  as  to  relieve  this  difHculty,  others  still  remain. 

Kant's  theory  has  also  secondary  difficulties  of  a  serious 
nature.  For  example,  in  a  gas  the  mutual  attractions  of 
the  molecules  could  not  draw  them  together  into  small  nuclei. 
Even  the  moon  could  not  now  add  to  its  mass  if  it  should 
pass  through  a  gas.  To  avoid  this  difficulty  one  might  assume 
that  there  was  first  condensation  into  the  liquid  or  solid  state. 
So  many  molecules  would  be  involved  in  the  formation  of 
even  the  minutest  drop  that,  by  an  averaging  process,  their 
lateral  motions  would  essentially  destroy  one  another,  the 
particle  would  fall  toward  the  center  of  the  whole  system, 
and  no  planets  would  be  formed.  In  order  to  avoid  this 
difficulty  it  is  necessary  to  depart  from  Kant's  ideas  and  to 
assume  either  that  the  whole  gaseous  mass  was  rotating  with 
considerable  velocity,  or  that  the  matter  was  not  in  a  gaseous 
state.  If  the  first  of  the  two  assumptions  is  made,  it  is  found 
by  a  mathematical  treatment  that  the  moment  of  momen- 
tum of  the  system  would  be  more  than  200  times  what  it  is 
at  present.  Since  the  moment  of  momentum  would  remain 
unaltered,  the  second  alternative  must  be  adopted.  But 
this  is  directly  contrary  to  the  fundamental  assumptions  of 
Kant,  and  it  is  hardly  permissible  to  regard  a  theory  as  hav- 
ing preserved  its  identity  after  having  been  modified  to  this 
extent.  The  condition  to  which  one  is  forced,  viz.,  that  of 
discrete  particles  in  orbital  revolution  in  the  same  direction, 
is  actually  the  planetesimal  organization. 

In  successive  chapters  Kant  considered  the  densities  and 


CH.  xn,  262]  EVOLUTION  OF  THE   SOLAR  SYSTEM     449 


ratios  of  the  masses  of  the  planets,  the  eccentricities  of  the 
planetary  orbits  and  the  origin  of  comets,  the  origin  of  satel- 
lites and  the  rotation  of  the  planets,  etc.  He  even  claimed 
to  have  proved  without  observational  evidence  the  existence 
of  life  on  other  planets.  In  spite  of  the  keenness  of  his 
intellect  and  his  remarkable  powers  of  generaUzation,  his 
theory  has  not  had  much  influence  on  speculations  in  cos- 
mogony, because  it  is  marred  by  so  many  serious  errors  in 
the  apphcation  of  physical  and  dynamical  laws. 

262.  The  Hjrpothesis  of  Laplace.  —  The  hypothesis  of 
Laplace  appeared  near  the  end  of  a  splendid  popular  work 
on  astronomy  which  he  pubhshed 
in  1796.  He  advanced  it  "  with 
that  distrust  which  everything 
ought  to  inspire  that  is  not  a 
result  of  observation  or  of  cal- 
culation." How  great  an  advance 
over  Kant  this  one  sentence 
indicates  ! 

In  outline,  the  hypothesis  of 
Laplace  was  that  originally  the 
solar  atmosphere  (in  later  edi- 
tions  a  nebulous  envelope)  ,_in  an 
intensely  heated  state,  extended 
out  beyond  ^the  orbit_  of_  the^ 
farthest  planeF;  the  whole  mass  fig.  162.  —  Laplace. 

rotated  as  a  solid  in  the  direc- 
tion in  which  the  planets  now  revolve ;  the  dimensions  of 
the  solar  atmosphere  were  maintained  mostly  by  gaseous 
expansion  of  the  highly  heated  vapors,  and  only  slightly 
by  the  centrifugal  acceleration  due  to  the  rotation ;  as  the 
mass  lost  heat  by  radiatixin,  it  jjontracted  under  the  mutual 
gravitation  of  its  parts ;_  simultaneously..  wittLlt^^  con- 
traction,  its  rate  oTl-otajjgn_  necj^ajriJxJngreagedL^ecause 
the  moment,  of  momentum  remained  constant ;  after  the 
rotating  mass  had  contracted  to  certain  dimensions  the  cen- 
2o 


450     AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xu,  262 

trif  ugal  acceleration  at  the  equator  equaled  the  attraction  by 
the  interior  parts  and  an  equatorial  ring  was  left  behind,  the 
remainder  continuing  to  contract ;  a  ring  was  abandoped 
at  the  distance  of  each  £lanetjL..a  ring  coiLld_^^  have 

had  absolute  uniformity,  and,  separating  at  some  point,  it 
united  at  some  other  because  of  the  mutual  attractions  of 
its  parts  and  formed  a  planet ;  and,  finally,  the  satellites  were 
formed  from  rings  which  were  left  off  by  the  contracting  plan- 
ets, Saturn's  rings  being  the  only  examples  still  remaining. 

The  contraction  theory  of  the  sun's  heat,  which  was  devel- 
oped by  Helmholtz  in  1854,  fitted  in  very  well  with  the  La- 
placian  hypothesis  and  was  considered  as  supporting  it. 
Some  objections  to  the  Laplacian  theory,  however,  began  to 
appear.  In  1873  Roche,  the  author  of  the  theorem  that  a 
satellite  would  be  broken  up  by  tidal  strains  if  its  distance 
from  its  primary  should  become  less  than  2.44  radii  of  the 
latter,  seriously  undertook  to  modify  the  hypothesis  of 
Laplace  so  as  to  relieve  it  of  the  difficulties  with  which  it 
was  beset.  His  modifications  were  for  the  most  part  improb- 
able and  do  not  in  the  least  meet  later  objections.  Kirk- 
wood,  an  American  astronomer,  criticized  the  Laplacian 
hypothesis  and  poin,ted  out  that  the  direct  rotation  of  the 
planets  might  be  due  to  the  effect  of  the  sun's  tides  on  them 
when  they  were  expanded  in  the  gaseous  state.  In  1884 
Faye  made  very  radical  modifications  of  the  hypothesis  of 
Laplace  for  the  purpose  of  avoiding  the  difficulties  in  which 
it  was  becoming  involved.  He  supposed  that  the  planets 
were  formed  in  the  depths  of  the  solar  nebula  and  that  those 
nearer  the  sun  are  actually  older  than  those  which  are  more 
remote.  About  1878  Darwin  began  his  great  work  on  the 
tides  which  he  regarded  as  supplementing  and  strengthening 
the  hypothesis  of  Laplace. 

It  is  now  generally  recognized  that  the  Laplacian  hypothec 
sis  fails  because  it  does  not  meet  the  most  fundamental 
requirements  of  the  problem.  For  example,  the  density  of 
the  hypothetical  solar  atmosphere  must  have  varied  in  har- 


CH.  XII,  262]   EVOLUTION  OF   THE    SOLAR   SYSTEM     451 

mony  with  the  laws  of  gases.  With  this  distribution  of 
density,  which  can  be  theoretically  determined,  and  the 
rotation  which  is  given  by  the  revolution  of  the  planets,  it  is 
an  easy  matter  to  compute  the  moment  of  momentum  pos- 
sessed  by  the  hypothetical  system  when  it  extended  out  to 
the  orbit  of  Neptune.  It  turns  out  to  be  more  than  200 
times  that  of  the  system  at  present.  If  the  hypothesis  of 
Laplace  were  correct,  the  two  would  be  equal ;  the  discrep- 
ancy is  so  enormous  that  the  hypothesis  must  be  radically 
wrong. 

The  details  of  the  Laplacian  hypothesis  are  subject  tcP 
equally  serious  difficulties.  For  example,  it  would  be  impos- 
sible  for  successive  ringsJaJje  left  off-  Kirkwood  long  ago 
pointed  out  that  if  instability  in  the  equatorial  zone  once 
set  in,  it  would  persist,  and  Chamberlin  has  shown  that  the 
result  would  be  a  continuous  disk  of  particles  describing 
nearly  circular  orbits.  Further,  if  a  ring  were  left  off,  it 
could  not  even  begin  to  cojadense  .intcL  a  planet  becausBJ>oth 
gaseous  expansion  and  the  tidal  forces  due  to  the  sun  would 
more  than  offset  the  mutual  gravitation  of  its  parte.  It  has 
been  seen  how  large  and  dense '  a  planet  must  be  in  order 
to  hold  an  atmosphere;  while  the  ring  would  be  large,  its 
density  would  be  extremely  low  and  it  could  not  control  the 
Ughter  elements.  And  it  has  been  shown  that  even  if  a  cir- 
cular ring  had  in  some  way  largely  condensed  into  a  planet, 
the  process~could  not" have  completed  itsilE  In  order  that 
a  nucleus  may  gather  up  scattered  materials,  it  is  necessaiy 
that  they  shall  be  moving  ianonsiderablv-eccentric  orbits. 

Since  the  Laplacian  hypothesis  fails  in  the  fundamental 
requirement  of  moment  of  momentum,  as  well  as  in  a  num- 
ber of  other  essential  respects,  it  will  be  sufficient  simply  to 
enumerate  some  of  the  phenomena  which  are  obviously  not 
in  harmony  with  it : 

(1)    It  does  not  provide  for  the  planetoids  with  their 

1  The  power  of  control  of  a  planet  on  an  atmosphere  is  proportional  to 
the  product  of  its  density  and  radius. 


452     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  m^ 

interlacing  orbits,  some  having  high  inclinations  or  eccen- 
tricities. 

(2)  It  does  not  permit  of  the  existence  of  an  object  hav- 
ing such  an  orbit  as  that  of  Eros,  which  reaches  from  near 
that  of  the  earth  out  beyond  that  of  Mars. 

(3)  It  impUes  that  a  continuous  disk  of  particles,  such  as 
that  producing  the  zodiacal  light,  cannot  exist. 

(4)  It  does  not  anticipate  the  considerable  eccentricity 
and  inclination  of  Merciu'y's  orbit. 

(5)  It  does  not  agree  with  the  fact  that  the  terrestrial 
planets  seem  to  be  at  least  as  old  as  the  more  remote  ones. 

(6)  It  does  not  permit  of  there  being  any  retrograde  satel- 
htes  because  the  rings  abandoned  by  a  contracting  nebula 
would  necessarily  all  rotate  in  the  same  direction.' 

(7)  It  impUes  that  the  rotation  period  of  each  planet  shall 
be  shorter  than  the  shortest  period  of  revolution  of  its  satel- 
Utes.  This  condition  is  not  only  violated  in  the  case  of  the 
inner  satelhte  of  Mars,  but  the  particles  of  the  inner  ring  of 
Saturn  revolve  in  half  the  period  of  the  planet's  rotation. 

263.  Tidal  Forces.  —  The  sun  and  moon  generate  tides 
in  the  oceans  that  cover  the  earth.  Tides  are  raised  also 
in  the  atmosphere  and  in  the  soHd  earth  itself.  Similarly, 
every  celestial  body  raises  tides  in  every  other  celestial  body. 
The  first  problem  which  will  be  considered  here  will  be  the 
character  of  the  tide-raising  forces,  and  the  second  will  be 
the  effects  of  the  tides  on  the  rotations  and  revolution  of  the 
two  bodies. 

Consider  the  tide-raising  effects  of  m  on  M,  Fig.  163. 
For  simphcity,  consider  the  effects  of  m  on  P  and  P',  two 
particles  on  the  surface  of  M.  The  problem  of  the  resolu- 
tion of  the  forces  is  that  which  was  treated  in  Art.  153.  Let 
MA  represent  the  acceleration  of  m  on  M  in  direction  and 
amount.  Then  the  acceleration  of  m  on  P  and  P'  will  be 
represented  by  PB  and  P'B'  respectively.     The  former  is 

1  Attempts  have  been  made,  though  not  successfully,  to  avoid  this  diffi- 
culty by  invoking  tidal  friction  (Art.  264). 


CH.  XII,  263]   EVOLUTION  OF   THE   SOLAR   SYSTEM     453 

greater  than  MA  because  the  acceleration  varies  inversely 
as  the  square  of  the  distance,  and  Mm  is  greater  than  Pm. 
For  a  similar  reason  P'B'  is  less  than  MA.  Now  resolve 
PB  into  two  components,  PC  and  PD,  in  such  a  way  that 
PC  shall  be  equal  and  parallel  to  MA.  Similarly,  resolve 
P'B'  into  P'C,  equal  and  parallel  to  MA,  and  P'D'.  Since 
PC  and  P'C  are  equal  and  parallel  to  MA,  they  have  no 
tendency  to  displace  P  and  P'  respectively  with  respect  to 
M.  The  remaining  components,  PD  and  P'D',  are  the  tide- 
raising  accelerations. 

Now  consider  the  tide-raising  forces  all  around  M.    They 
are  as  indicated  in  Fig.  94.     The  forces  toward  m  are  slightly 


r:-.>=-o 


Fig.  163.  —  The  tide-raising  force. 

greater  than  those  in  the  opposite  direction,  while  the  com- 
pressional  forces  at  90°  from  these  directions  are  half  as  great. 
It  is  clear  from  this  figure  that  if  M  were  not  rotating  and  m 
were  not  revolving  around  it,  there  would  be  a  tide  on  the 
side  of  M  towards  m,  and  a  nearly  equal  tide  on  the  side  of 
M  away  from  m  (compare  Art.  153).  The  motions  of  the 
bodies  produce  important  modifications. 

Suppose  the  rotation  of  M  and  the  revolution  of  m  are  in 
the  same  direction  and  that  the  period  of  rotation  of  M  is 
shorter  than  that  of  the  revolution  of  m.  This  is  the  case 
in  the  earth-moon  system.  Under  these  circumstances  the 
tides  Ti  and  Tj  are  carried  somewhat  ahead  of  the  line  Mm, 
as  represented  in  Fig.  164.  The  more  nearly  equal  the  rates 
of  rotation  of  M  and  revolution  of  m,  the  more  nearly  will 
the  tides  Ti  and  T2  be  in  the  line  Mm. 


454     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  263 

Consider  a  point  on  the  rotating  body  M.  It  will  first 
pass  the  Une  Mm,  and  then  somewhat  later  it  will  pass  the 
tide  Ti.  The  interval  is  the  lag  of  the  tide.  In  the  case 
of  the  earth-moon  system  a  meridian  passes  eastward  across 
the  moon  (the  moon  seems  to  pass  westward  across  the 
meridian),  and  somewhat  later  the  meridian  passes  the  tidal 
cone  and  has  high  tide.  The  problem  is  enormously  compli- 
cated in  the  case  of  the  earth  by  the  addition  of  the  tides  due 
to  the  sun,  by  the  varying  distances  of  the  moon  and  sun 
north  or  south  of  the  celestial  equator,  by  their  changing  dis- 
tances from  the  earth,  and  especially  by  the  irregular  con- 
tours of  the  continents  and  the  varjdng  depths  of  the  oceans. 
These  modifying  factors  are  so  numerous  and  in  some  cases 
so  poorly  known  that  at  present  it  is  not  possible  to  predict 
entirely  in  advance  of  observation  either  the  times  or  heights 
of  the  tides ;  but,  after  a  few  observational  data  have  estab- 
lished the  way  in  which  the  tides  at  a  station  depend  upon 
the  unknown  factors,  predictions  become  thoroughly  rehable, 
for  the  tides  vary  in  perfect  harmony  with  the  tidal  forces. 

264.  Tidal  Evolution.  —  The  tides  are  not  fixed  on  the 
surface  of  M,  Fig.  164,  unless  the  period  of  its  rotation  equals 
the  period  of  revolution  of  m.  When  the  periods  are  unequal 
so  that  the  tides  move  around  the  rotating  body,  some  energy 
is  changed  to  heat  by  friction.  This  energy  comes  from  the 
kinetic  and  potential  energies  of  the  system ;  and,  in  accord- 
ance with  the  laws  of  dynamics,  the  transformation  of 
energy  takes  place  in  such  a  way  that  the  total  moment  of 
momentum  of  the  system  remains  unchanged.  Of  course, 
in  general  there  will  be  tides  on  both  of  the  mutually  attract- 
ing bodies. 

The  character  of  the  transformation  of  energy  that  takes 
place  under  tidal  friction  depends  upon  the  dynamical 
properties  of  the  system.  Suppose  that  the  motions  of 
rotation  and  revolution  are  in  the  same  direction  and  that 
the  period  of  M  is  shorter  than  that  of  m.  It  can  be  shown 
that  under  these  circumstances  the  periods  of  both  M  and  m 


CH.  XII,  264]  EVOLUTION  OP  THE   SOLAR  SYSTEM     455 

and  their  distance  apart  are  increased.  The  reason  that  the 
period  of  rotation  of  M  is  increased  is  that  m  has  a  component 
of  attraction  back  on  both  Ti  and  T^,  Fig.  164,  as  can  be 
shown  by  resolving  the  forces  as  they  were  resolved  in  Fig. 
163.  If  m  pulls  Ti  and  Ta  backward,  it  follows  from  the 
reaction  of  forces  that  Ti  and  T^  pull  m  forward.  The  result 
of  a  forward  component  on  m  is  to  increase  the  size  of  its 
orbit  and  to  lengthen  its  period. 

If  m  is  near  M,  the  effect  of  the  tides  on  the  period  of  revo- 
lution of  m  is  greater  than  their  effect  on  the  period  of  rotation 
of  M.    If  the  bodies  are  far  apart,  the  result  is  the  opposite. 

Suppose  the  two  bodies  are  initially  close  together  and  that 
the  period  of  rotation  of  M  is  only  a  little  shorter  than  the 


-o 


Fig.  164.  —  Tidal  cones  and  the  lag  of  the  tides. 

period  of  revolution  of  m.  The  friction  of  the  tides  will 
lengthen  both  periods  and  increase  the  difference  between 
them.  If  nothing  else  interferes,  this  will  continue  until  a 
certain  distance  between  the  bodies  has  been  reached.  After 
that,  the  effect  on  the  period  of  rotation  of  M  will  be  greater 
than  that  on  the  period  of  revolution  of  m.  Consequently, 
although  both  periods  will  continue  to  increase  in  length, 
they  will  approach  equality.  Eventually,  the  periods  of 
rotation  and  revolution  will  be  equal,  the  tides  will  remain 
fixed  on  M,  and  there  will  be  no  further  tidal  friction  or 
tidal  evolution. 

The  most  important  case  frpm  a  practical  point  of  view 
has  been  considered,  but  there  are  two  others.  In  the  first 
the  bodies  move  in  the  same  direction,  but  the  period  ,of 


456     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  264 

rotation  of  M  is  longer  than  that  of  revolution  of  m.  Under 
these  circumstances  both  periods  are  decreased,  the  relative 
amounts  depending  on  the  distance  of  the  bodies  from  each 
other.  If  the  bodies  are  initially  far  apart,  the  effect  will  be 
greater  on  the  period  of  rotation  of  M  than  on  the  period  of 
revolution  of  m,  and  the  two  periods  will  approach  equaUty. 
But  if  the  bodies  are  near  together,  the  effect  will  be  relatively 
greater  on  the  period  of  m,  the  periods  will  not  approach 
equality,  and  the  bodies  wiU  ultimately  collide.  In  the 
second  case  the  rotation  of  M  is  in  the  direction  opposite  to 
that  of  the  revolution  of  m.  Under  these  circmnstances 
M  rotates  faster  and  faster,  the  distance  of  m  continually 
decreases,  and  the  inevitable  outcome  is  the  collision  and 
union  of  the  two  bodies. 

The  rate  at  which  tidal  friction  takes  place  depends  upon 
the  physical  properties  of  the  bodies.  If  they  are  perfect 
fluids  so  that  there  is  no  degeneration  of  energy,  there  is  no 
tidal  evolution.  Likewise  if  they  are  perfectly  elastic,  there 
is  no  tidal  friction. 

The  rate  of  tidal  friction  also  depends  upon  the  difference 
in  the  periods  of  the  two  bodies.  If  the  difference  between 
the  periods  is  small,  the  tides  Ti  and  T2,  Fig.  164,  are  almost 
in  the  line  Mm,  and  it  is  obvious  that  the  backward  compo- 
nents are  small  and  the  rate  of  tidal  friction  is  very  slow. 
Suppose  the  periods  are  approaching  equality.  The  smaller 
their  difference  the  slower  is  their  rate  of  change,  and  they 
never  become  exactly  equal  but  approach  equality  as  the 
time  becomes  infinitely  great. 

265.  Effects  of  the  Tides  on  the  Motions  of  the  Moon.  — 
The  most  striking  thing  in  the  earth-moon  system  is  that 
the  moon's  periods  of  rotation  and  revolution  are  equal. 
It  is  extremely  improbable  that  this  unique  relation  is  acci- 
dental. The  only  explanation  of  it  heretofore  advanced  is 
that  the  moon's  period  of  rotation  has  been  brought  into 
equality  with  its  period  of  revolution  by  the  tides  generated 
in  it  by  the  earth. 


CH.  XII,  265]   EVOLUTION  OF   THE    SOLAR   SYSTEM     457 

The  tidal  force  exerted  by  the  earth  on  the  moon  is  about 
20  times  the  tidal  force  exerted  by  the  moon  on  the  earth. 
The  amount  of  tidal  friction  is  proportional  to  the  square  of 
the  tidal  force.  Therefore,  if  the  physical  properties  of  the 
earth  and  moon  were  the  same  and  if  their  periods  of  rotation 
were  equal,  the  effectiveness  of  the  tides  generated  by  the 
earth  on  the  moon  in  changing  the  moment  of  momentum 
of  the  moon  would  be  400  times  that  of  the  tides  generated 
by  the  moon  on  the  earth  in  changing  the  moment  of  momen- 
tum of  the  earth .  Since  the  moment  of  momentum  of  a  body 
is  proportional  to  the  product  of  its  mass  and  the  square  of  its 
radius,  a  given  change  in  the  moment  of  momentum  of  the 
moon  alters  its  rate  of  rotation  1200  times  as  much  as  the 
same  change  in  moment  of  momentum  alters  the  rate  of 
rotation  of  the  earth.  Consequently,  taking  the  two  fac- 
tors together,  if  the  earth  and  moon  were  physically  ahke 
and  had  the  same  period  of  rotation,  tidal  friction  would 
change  the  period  of  rotation  of  the  moon  400  X  1200  = 
480,000  times  as  fast  as  it  would  change  the  period  of  rotation 
of  the  earth. 

The  results  which  have  been  obtained  seem  to  be  very 
favorable  to  the  theory  that  the  tides  have  caused  the 
moon  to  present  one  side  toward  the  earth,  but  some  serious 
difficulties  remain.  It  can  be  shown  that,  considering  the 
tidal  interactions  of  the  earth  and  moon  and  the  effect  of 
the  sun's  tides  on  the  moon,  the  present  condition  of  the 
earth-moon  system  is  not  one  of  equilibrium.  The  tides 
raised  by  the  earth  on  the  moon  have  no  effect  under  present 
circumstances  on  the  rotation  and  revolution  of  the  moon. 
The  tides  raised.by  the  moon  on  the  earth  increase  the  length 
of  the  month  but  do  not  affect  the  rotation  of  the  moon. 
The  tides  raised  by  the  sun  on  the  moon  increase  the  moon's 
period  of  rotation  but  do  not  affect  its  revolution.  Conse- 
quently the  moon's  periods  of  rotation  and  revolution  are 
both  increasing,  and  it  is  infinitely  improbable  that  all  the 
factors  on  which  these  effects  depend  are  so  related  that 


458     AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xii,  265 

they  are  exactly  equal.  Even  if  they  were  equal  at  one  time, 
they  would  become  unequal  with  a  changed  distance  of  the 
moon  from  the  earth.  That  is,  the  present  is  not  a  fixed  state 
of  equilibrium,  and  the  consideration  of  the  tides  does  not 
remove  the  difficulties.  It  seems  probable  from  this  line  of 
thought  that  some  influence  so  far  not  considered  has  caused 
the  moon  always  to  present  the  same  face  toward  the  earth. 

266.  Effects  of  the  Tides  on  the  Motions  of  the  Earth.  — 
The  theory  of  the  tidal  evolution  of  the  earth-moon  system, 
on  the  basis  of  certain  assumptions  regarding  the  physical 
condition  of  the  earth,  was  elaborated  by  Sir  George  Darwin 
in  a  splendid  series  of  investigations.  While  the  experiment 
of  Michelson  and  Gale  (Art.  25)  proves  that  his  assumptions 
are  not  satisfied,  at  least  at  the  present  time,  the  possible 
sequence  of  events  which  he  worked  out  is  interesting. 

Since  the  tides  are  increasing  the  lengths  of  both  the 
day  and  the  month,  both  of  these  periods  were  formerly 
shorter  and  the  moon  was  nearer  the  earth.  On  the  basis 
of  his  assumptions,  Darwin  traced  the  day  back  until  it  was 
only  four  or  five  of  our  present  hours.  At  that  time  the 
moon  was  revolving  close  to  the  earth  in  a  period  almost 
equally  short.  This  led  him  to  the  conclusion  that  at  an 
earher  stage  the  earth  and  moon  were  one  body,  that  they 
divided  into  two  parts  because  of  the  rapid  rotation  of  the 
combined  mass,  and  that  they  have  attained  their  present 
state  as  a  consequence  of  tidal  friction.  The  same  reason- 
ing leads  to  the  conclusion  that  in  the  future  they  will  con- 
tinue to  separate  and  that  the  day  will  continually  increase 
in  length. 

The  critical  question  is  whether  the  physical  properties 
of  the  earth  are  such  that  the  rate  at  which  tidal  evolution 
takes  place  makes  it  an  appreciable  factor  in  the  history  of 
the  earth.  Darwin  supposed  the  main  effects  were  due  to 
bodily  tides  in  the  earth  which  he  assumed  to  be  viscous. 
Since  it  is  highly  elastic,  they  cannot  at  present  be  important, 
but  it  has  generally  been  assumed  that,  whatever  its  present 


CH.  XII,  266]   EVOLUTION  OF   THE   SOLAR  SYSTEM     459 

condition  may  be,  it  was  formerly  viscous.  There  is  abso- 
lutely no  evidence  to  support  the  assumption,  and  if  its 
present  properties  of  solidity  and  elasticity  are  a  conse- 
quence of  the  pressure  in  its  interior,  the  assumption  seeiris 
very  improbable.  As  Poisson  and  Lord  Kelvin  showed, 
the  temperature  of  the  interior  of  the  earth  cannot  have 
fallen  appreciably  in  several  hundreds  of  millions  of  years  by 
the  conduction  of  heat  to  its  surface.  Since  the  tempera- 
ture, of  the  interior  of  the  earth  cannot  have  changed  appre- 
ciably, there  seems  to  be  no  good  ground  for  assuming  that 
the  earth  was  once  viscous. 

Since  there  cannot  now  be  an  important  degeneration  of 
energy  in  the  bodily  tides  of  the  earth,  tidal  evolution  must 
depend  at  present  almost  entirely  upon  the  tides  in  the 
ocean  and  the  atmosphere.  The  latter  may  be  neglected 
without  important  error.  The  oceanic  tides  are  so  irregular 
that  it  is  difficult  to  determine  their  effects  on  the  rotation 
of  the  earth.  But  MacMillan  has  made  liberal  estimates  of 
the  unknown  factors,  and  has  found  that  at  present  the 
length  of  the  day  cannot  be  increasing  at  a  rate  of  more 
than  one  minute  in  30,000,000  years. 

It  is  possible  to  determine  observationally  the  present  rate 
of  tidal  evolution.  The  day  and  the  month  are  increasing 
in  length,  but  the  effect  on  the  day  iff  the  greater.  Conse- 
quently, if  the  length  of  the  month  is  measured  in  days,  as 
is  done  practically,  it  will  seem  to  be  decreasing  in  length. 
It  is  found  from  observations  that  the  moon  is  getting  ahead 
of  its  predicted  place  from  4  to  6  seconds  of  arc  in  100  years. 
That  is,  in  1240  revolutions  the  moon  gets  ahead  of  its  pre- 
dicted place  about  -^  of  its  diameter.  On  the  basis  of  these 
facts  and  the  assumption  that  the  increase  in  the  length  of 
the  month  is  due  to  the  tidal  interactions  of  the  earth  and 
moon,  the  proper  discussion  shows  that  at  the  present  time 
the  length  of  the  day  is  increasing  as  a  consequence  of  all  the 
factors  affecting  the  rotation  of  the  earth  at  the  rate  of  one 
minute  in  900,000,000  years. 


460     AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xii,  266 

It  is  evident  th^t  tidal  evolution  is  not  an  important  fac- 
tor in  the  earth-moon  system  f of  any  period  short  of  several 
hundred  millions  of  years.  Either  the  theory  of  tidal  evolu- 
tion as  elaborated  by  Darwin  must  be  abandoned  as  not 
representing  what  has  actually  taken  place,  or  a  condition 
for  the  earth's  interior  totally  different  from  that  which  exists 
at  present  must  be  arbitrarily  assumed. 

267.  Tidal  Evolution  of  the  Planets.  —  There  is  perhaps 
some  slight  evidence  that  Mercury  and  Venus  always  keep 
the  same  side  toward  the  sun,  and  this  condition  has  been 
ascribed  to  the  effects  of  tides  which  the  sun  may  have  raised 
in  them.  The  tidal  force  exerted  by  the  sun  on  Mercury  is 
about  2.5  times  as  great  as  that  of  the  moon  on  the  earth. 
In  view  of  the  fact  that  the  moon's  tides  on  the  earth  cer- 
tainly do  not  have  appreciable  effects,  it  does  not  seem  prob- 
able that  the  sun's  tides  have  radically  changed  the  rotations 
of  Mercury  and  Venus.  Besides  this,  it  must  be  remem- 
bered that  the  condition  of  equality  of  periods  of  rotation  and 
revolution  would  be  attained  in  any  case  only  after  an 
infinite  time. 

The  tidal  action  of  the  sun  on  the  more  remote  planets  is 
much  less  than  that  on  the  earth.  No  other  satellite  has 
relatively  as  great  effects  on  its  primary  as  the  moon  has  on 
the  earth.  Consequently,  we  are  forced  to  the  conclusion 
that  in  the  solar  system  tidal  evolution  has  not  been  an  im- 
portant factor  for  a  period  of  at  least  several  hundreds  of 
milhons  of  years. 

XXI.    QUESTIONS  ' 

1.  According  to  Kant's  theory,  why  should  the  sun  rotate  in  the 
direction  the  planets  revolve  ? 

2.  Is  the  assumption  of  Laplace  that  the  original  nebula  was 
highly  heated  in  harmony  with  the  present  temperature  of  the  sun 
and  Lane's  law?    Why  did  Laplace  make  the  assumption? 

3.  Why  did  Laplace  assume  that  the  original  nebula  was  rotating 
as  a  sohd  ? 

4.  To  what  extent  does  the  contraction  theory  of  the  sun's  heat 


CH.  XII,  267]   EVOLUTION  OF  THE   SOLAR  SYSTEM     461 

support  the  Laplacian  hypothesis?     Is  it  opposed  to  the  planetes- 
imal  hypothesis  and  Kant's  hypothesis  ? 

5.  In  what  way  does  the  Laplacian  hypothesis  fail  to  meet  the 
requirements  of  moment  of  momentum  ? 

6.  On  the  basis  of  Lane's  law,  what  was  the  temperature  of  the 
surface  of  the  sun  if  it  extended  to  the  orbit  of  the  earth  ?  How  do 
you  account  for  the  presence  of  refractory  materials  in  the  earth, 
under  the  Laplacian  hypothesis  ? 

7.  Explain  carefully  in  what  respects  the  seven  things  mentioned 
at  the  end  of  Art.  262  are  opposed  to  the  Laplacian  hypothesis. 

8.  What  should  be  the  present  shape  of  the  sun  if  the  Laplacian 
hypothesis  were  true  ? 

9.  In  the  case  of  the  earth  and  moon,  what  is  the  magnitude  of 
the  tidal  force  at  the  point  on  the  side  of  the  earth  toward  the  moon 
compared  to  the  whole  attraction  of  the  moon?  Compared  to  the 
attraction  of  the  earth? 

10.  The  tides  produced  on  the  earth  by  the  moon  decrease  the 
earth's  moment  of  momentum;  what  becomes  of  that  which  the 
earth  loses,  and  what  changes  in  the  system  does  it  cause  ? 

11.  Show  that  when  M  rotates  faster  than  m  revolves,  the 
attractions  of  m  for  both  Ti  and  Tj  tend  to  decrease  the  rate  of 
rotation  of  M. 

12.  Suppose  the  rate  of  rotation  of  the  earth  is  constant  and  that 
in  a  century  the  moon  gets  6"  ahead  of  the  place  it  would  occupy 
if  its  rate  of  revolution  were  constant.  How  long  would  be  required 
for  its  period  to  decrease  1  per  cent  ? 


462    AN  INTRODUCTION  TO  ASTRONOMY    [ch.  xii,  20/ 


Fig.   165.  —  Milky  Way  in  Aquila.     Photographed  by  Barnard  at  the  Yerkes 
Observatory,  August  27,  1906. 


CHAPTER  XIIi; 
THE    SIDEREAL   UNIVERSE 

I.   The  Apparent  Disthibution  of  the  Staes 

268.   On  the  Problems  of  the  Sidereal  Universe.  —  The 

invention  of  the  telescope  and  the  discovery  of  the  law  of 
gravitation  were  followed  by  a  long  period  of  successes  in 
unraveling  the  mysteries  of  the  solar  system-.  •  The  positions 
of  the  sun,  moon,  and  planets  were  measured  with  ex- 
traordinary precision,  and  the  law  of  gravitation  accounted 
for  the  nmnerous  pecuUarities  of  their  motions.  What  had 
been  mysterious  and  inexpUcable  became  famihar  and  thor- 
oughly imderstood.  Telescopes  of  continually  increasing 
power  enabled  astronomers  to  measure  accurately  the  dis- 
tances and  diameters  of  these  bodies  and  to  learn  much  of 
their  surface  conditions.  At  last  the  invention  of  the  spec- 
troscope enabled  them  to  determine  the  chemical  constitution 
of  the  sun. 

There  is  great  pleasure  now  in  working  in  a  science  whose 
data  are  exact  and  whose  laws  are  firmly  established.  The 
certainty  of  the  results  satisfies  the  human  instinct  for  final 
truth.  But  there  was  also  pleasure  of  a  different  kind  for 
those  pioneers  who  first  explored  the  planetary  system  with 
good  instrmnents  and  showed  "by  mathematical  processes 
that  its  members  are  obedient  to  law.  For  them  every 
observation  and  every  calculation  was  an  adventure.  They 
were  continually  in  fear  that  their  dreams  of  knowing  the 
order  prevailing  in  the  universe  would  be  shattered;  they 
were  continually  elated  by  having  their  theories  confirmed. 

The  pioneer  days  of  discovery  in  the  solar  system  are  past. 
Not  that  great  discoveries  do  not  remain  to  be  made,  but 

463 


464    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  268 

they  will  henceforth  fit  into  a  large  body  of  organized  facts. 
From  now  on  the  romance  and  excitement  of  astronomical 
adventure  will  be  largely  furnished  by  the  explorations  of  the 
sidereal  universe.  Astronomers  have  become  accustomed 
to  the  fact  that  the  sun  is  a  million  times  as  large  as  the  earth, 
and  famiUarity  has  dulled  their  amazement  at  its  terrific 
activities;  from  now  on  they  must  deal  with  millions  of 
stars,  some  of  them  much  larger  and  thousands  of  times 
more  luminous  than  the  sun.  They  have  measured  and  at 
least  partially  grasped  the  enormous  dimensions  of  the  solar 
system ;  from  now  on  they  must  conceive  of  and  deal  with 
distances  milUons  of  times  as  great.  They  have  observed 
the  differences  in  characteristics  exhibited  by  eight  planets ; 
from  now  on  they  will  be  face  to  face  with  the  infinite  diver- 
sity presented  by  the  stars.  They  have  definitely  accepted 
the  doctrine  that  the  solar  system  has  undergone  a  great 
evolution  whose  details  are  yet  much  in  doubt;  the  corre- 
sponding question  for  hundreds  of  milUons  of  other  systems 
is  looming  up  more  indistinctly  through  the  fogs  of  uncer- 
tainties which  still  surround  them.  It  might  be  supposed 
that  astronomers  would  be  tempted  to  lay  down  the  arms 
of  their  understanding  before  the  transcendental  and  appal- 
lingly difficult  problems  presented  by  the  sidereal  system. 
Instead,  with  all  the  weapons  at  their  command,  they  are 
making  more  vigorous,  persistent,  and  successful  attacks  than 
ever'  before.  Astronomers  of  all  the  leading  countries  are 
united  and  cooperate  in  this  campkign;  they  employ  tele- 
scopes of  many  kinds,  spectroscopes,  photographic  plates, 
measuring  machines,  and  powerful  mathematical  processes  in 
their  attempts  to  penetrate  the  unknown. 

269.  The  Number  of  Stars  of  Various  Magnitudes.  — 
The  simplest  and  most  easily  determined  thing  about  the 
stars  is  their  number.  Of  course  the  number  that  can  be 
seen  depends  upon  the  power  of  the  instrument  with  which 
the  observations  are  made.  If  the  stars  extend  infinitely 
in  every  direction  with  approximately  equal  distances  from 


CH.  XIII,  269]        THE   SIDEREAL  UNIVERSE  465 

one  another,  the  number  of  them  revealed  by  a  telescope  will 
be  proportional  to  the  space  it  brings  within  visual  range. 
On  the  other  hand,  if  the  stars  are  less  densely  distributed  at 
a  great  distance  in  any  direction,  then  the  number  of  faint 
distant  stars  seen  in  that  direction  will  fall  short  of  being 
proportional  to  the  space  penetrated  by  the  instrument. 
For  this  reason  it  is  important  to  find  the  number  of  stars  of 
each'  magnitude  down  to  the  hmits  of  range  of  the  most 
powerful  telescopes. 

Consider  first  what  the  apparent  distribution  in  magnitude 
would  be  if  stars  of  every  actual  size  and  luminosity  were 
scattered  uniformly  throughout  space.  Take  a  large  enough 
volume  so  that  accidental  groupings  will  not  sensibly  affect 
the  results.  For  example,  suppose  there  are  5000  stars  in 
the  first  six  magnitudes  and  compute  the  number  there  should 
be,  under  the  hypothesis,  in  the  first  seven  magnitudes. 
The  sixth-magnitude  stars  are  2.512  •  •  •  times  as  bright 
as  the  seventh-magnitude  stars.  Since  the  magnitudes  of 
stars  of  any  given  absolute  brightness  are  directly  propor- 
tional to  the  squares  of  their  distances,  it  follows  that  stars 
of  the  seventh  magnitude  are  V2.512  •  •  •  =  1.585  •  •  •  times  as 
distant  as  corresponding  stars  of  the  sixth  magnitude. 
Therefore  the  volume  occupied  by  stars  out  to  the  seventh 
magnitude,  inclusive,  is  (1.585  •••)'=  3.98  •••  times  that 
occupied  by  the  first  six  magnitudes.  Hence,  if  the  stars 
were  uniformly  distributed  and  the  light  of  the  remote  ones 
were  in  no  way  obstructed,  there  would  be  3.98  •  •  •  times  as 
many  stars  in  the  first  seven  magnitudes  as  in  the  first  six 
magnitudes,  or  nearly  20,000  stars.  The  ratio  is  the  same 
for  the  total  number  of  stars  up  to  any  two  successive 
magnitudes  because  the  particular  magnitudes  do  not 
enter  into  its  computation.  And  it  can  be  shown  easily 
that  the  ratio  of  the  number  of  stars  of  any  magnitude  to  the 
number  of  the  next  magnitude  brighter  is  also  3.98  •  •  •. 

It  remains  to  examine  the  results  furnished  by  the  obser- 
vations. The  stars  are  so  extremely  numerous  that  only  a 
2h 


466    AN   INTRODUCTION   TO  ASTRONOMY   [ck.  xiii,  269 

small  fraction  of  the  total  number  within  reach  of  modern 
instruments  has  been  counted.  But  an  appi-oximation  to  the 
solution  of  the  problem  of  determining  the  number  of  stars 
has  been  obtained  by  coimting  sample  regions  of  known  size 
in  many  parts  of  the  sky,  and  then  multiplying  the  result 
by  the  number  necessary  to  include  the  whole  celestial  sphere. 
By  far  the  most  extensive  work  of  this  kind  has  been  carried 
out  by  Chapman  and  Melotte  of  the  Royal  Observatory  at 
Greenwich.  They  made  use  of  stars  down  to  magnitude 
17.5,  where  4,000,000  of  them  send  to  the  earth  only  a  little 
more  light  than  one  star  of  the  first  magnitude.  Their 
results  are  given  in  the  following  table :  ^ 

Table  XIV 


Magnitude 

Number  of  Stabs 

Maonitude 

NnMBER  op  Stabs 

5  to    6 

2,026 

11  to  12 

961,000 

6  to    7 

7,095 

12  to  13 

2,023,000 

7  to    8 

22,550 

13  to  14 

3,964,000 

8  to    9 

65,040 

14  to  15 

7,824,000 

9  to  10 

172,400 

15  to  16 

14,040,000 

10  to  11 

426,200 

16  to  17 

25,390,000 

The  ratio  of  the  number  of  stars  of  a  given  magnitude  to 
the  number  of  stars  one  magnitude  fainter  is  3.5  at  the 
beginning  of  the  table,  and  it  continually  decreases  to  1.8 
at  the  end.  Therefore,  not  only  is  the  ratio  for  every 
interval  of  one  magnitude  less  than  the  3.98  corresponding 
to  uniform  distribution  of  the  stars,  but  it  falls  off  about  50 
per  cent  in  12  magnitudes. 

What  conclusions  can  be  drawn  from  the  facts  given  by 
the  table  ?  It  is  certain  that  the  stars  cannot  be  uniformly 
distributed  to  indefinite  distances  unless  there  is  something 

'  The  juambers  in  the  first  of  this  table  disagree  with  those  in  Table  II 
because  here,  in  the  first  line,  for  example,  the  number  is  that  of  stars  from 
magnitude  5.0  to  6.0,  while  in  Table  II  the  corresponding  number  is  that  of 
stars  whose  magnitudes  are  4.5  to  5.5. 


CH.  XIII,  269]        THE    SIDEREAL  lllNIVERSE  467 

which  prevents  their  light  from  coming  to  us.  If  there  were 
a  sufficient  number  of  dark  stars  and  planets,  the  light  from 
remote  luminous  stars  would  be  shut  off ;  but  the  number  of 
non-luminous  bodies  required  to  account  for  the  black  sky- 
would  be  milHons  of  tipaes  the  nimiber  of  bright  ones.  In 
spite  of  the  fact  that  certain  variable  stars  (Art.  288)  prove 
the  existence  of  relatively  dark  bodies,  and  that  analogy 
with  the  planets  would  lead  to  the  conclusion  that  there 
are  many  non-luminous  bodies  of  secondary  dimensions, 
it  seems  extremely  improbable  that  they  are  sufficiently 
numerous  to  explain  the  observed  phenomena.  But  if  the 
obscure  matter  were  finely  divided,  as  in  meteoric  dust,  a 
given  mass  of  it  would  be  a  much  more  effective  screen,' 
and  the  total  mass  requirements  would  not  be  so  severe. 
Finely  divided  material  would  not  only  absorb  light,  but  it 
would  scatter  the  blue  light  and  cause  distant  stars  to  appear 
redder  than  nearer  stars  of  the  same  character. 

There  are  certain  phenomena  which  give  sHght  support 
to  the  hypothesis  that  there  is  some  scattering  of  fight  of 
this  nature,  but  they  are  not  conclusive.  One  of  them  is 
directly  related  to  the  question  in  hand.  Kapteyn  found 
from  an  investigation  of  stars  down  to  the  fourteenth  magni- 
tude, part  of  the  data  being  furnished  by  the  visual  obser- 
vations of  Sir  John  Herschel,  that  the  number  of  stars  of 
the  fainter  magnitudes  is  much  greater  than  is  given  in  the 
table  of  Chapman  and  Melotte.  The  faintest  stars  used  in 
the  construction  of  their  table  are  obtained  from  the  Franklin- 
Adams  photographic  charts  of  Greenwich.  Turner  has 
suggested  that,  because  of  the  scattering  of  fight,  the  remote 
faint  stars  may  be  deficient  in  the  blue  end  of  the  spectrum, 
to  which  photographic  plates  are  most  sensitive,  and  conse- 
quently that  a  considerable  part  of  the  stars  belonging  visu- 
ally to  a  certain  magnitude  belong  photographically  to  a 
fainter  magnitude.     In  spite  of  these  possible  indications  of 

1  The  effectiveness  of  opaque  matter  of  given  total  mass  in  cutting  off 
light  is  inversely  proportional  to  the  radius  of  its  separate  parts. 


468    AN   INTKODUCTION   TO  ASTRONOMY   [ch.  xiii,  i;69 

scattered  particles,  it  seems  extremely  improbable  that  the 
falling  off  of  the  star  ratio  from  3.98  to  1.8  is  due  appreciably 
to  this  cause. 

The  most  obvious,  though  not  necessary,  conclusion  which 
has  generally  been  drawn  from  the  table  is  that  the  stars  are 
hmited  in  number  and  that  they  occupy  a  hmited  portion  of 
space.  In  the  first  seventeen  magnitudes  there  are  in  round 
numbers  55,000,000  stars.  Chapman  and  Melotte  derived 
a  simple  formula  which  represented  the  numbers  closely 
for  these  magnitudes,  and  then,  under  the  assumption  that 
the  same  formula  holds  indefinitely  beyond,  they  deter- 
mined the  magnitude  for  which  there  are  as  many  stars 
brighter  as  there  are  fainter,  and  computed  the  total  mmiber 
of  stars  altogether.  By  this  process  they  concluded  that  the 
median  magnitude  hes  between  22.5  and  24.3,  which  are 
several  magnitudes  beyond  the  reach  of  existing  instru- 
ments, and  that  the  number  of  stars  of  all  magnitudes  is 
between  770,000,000  and  1,800,000,000.  It  is  obvious  that 
such  an  extrapolation  is  hazardous,  and  they  did  not  lay 
any  particular  stress  on  the  results.  In  fact,  the  data 
given  by  the  observations  can  be  as  exactly  represented  by 
many  other  less  simple  formulae  which  will  give  totally  dif- 
ferent results  for  the  fainter  magnitudes. 

There  is  an  even  simpler  line  of  reasoning  which  has  led 
many  astronomers  to  the  conclusion  that  the  material  imi- 
verse  is  hmited.  Since  the  stars  of  any  magnitude  are  2.512 
times  fainter  than  those  of  the  next  preceding  magnitude, 
and,  imder  the  hypothesis  of  uniform  distribution,  3.98 
times  more  numerous,  it  follows  that  if  the  star  density  did 
not  diminish  as  the  distance  increases,  the  stars  of  each 
magnitude  would  give  us  3.98  -;-  2.512  =  1.58  times  as  much 
light  as  those  of  the  next  magnitude  brighter.  Consequently, 
the  first  20  magnitudes  would  give  17,000  times  as  much  Ught 
as  the  first-magnitude  stars,  the  first  100  magnitudes  would 
give  168,000,000,000,000,000,000  times  as  much  light,  and  so 
on.    If  there  were  no  Umit  to  the  number  of  magnitudes  and 


CH.  XIII,  269]        THE   SIDEREAL  UNIVERSE  469 

no  absorbing  material,  there  would  be  no  limit,  except  for  the 
mutual  eclipsing  of  the  stars,  to  the  amount  of  light  received 
from  all  of  them.  The  sky  would  be  everywhere  ablaze 
with  the  average  brightness  of  a  star,  perhaps  equal  to  that 
of  the  sim.  The  stars  in  one  hemisphere  would  give  us  more 
than  90,000  times  as  much  light  as  the  sun,  but  actually 
the  sun  gives  us  15,000,000  times  as  much  light  as  all  the  stars 
together.  Therefore,  unless  much  Ught  is  absorbed,  the 
hypothesis  of  uniform  distribution  of  the  stars  to  infinity 
is  radically  false. 

Is  it  necessary,  therefore,  to  conclude  that  the  number  of 
stars  is  limited  and  that  they  occupy  only  a  finite  part  of 
space?  By  no  means;  simply  that  they  cannot  be  dis- 
tributed with  approximate  uniformity  throughout  infinite 
space.  It  was  pointed  out  by  Lambert  long  ago  that,  just 
as  the  solar  system  is  a  single  unit  in  a  galaxy  of  several  hun- 
dred million  stars,  so  the  Galaxy  may  be  but  a  single  one  out 
of  an  enormous  number  of  galaxies  separated  by  distances 
which  are  very  great  in  comparison  with  their  dimensions, 
and  that  these  galaxies  may  form  larger  units,  or  super- 
galaxies,  and  so  on  without  Umit.  There  is  nothing  in  such 
an  organization  which  is  inconsistent  with  the  facts  estab- 
Ushed  by  observation,  for  it  is  possible  to  build  up  infinite 
systems  of  stars  in  this  way  which  would  give  us  only  a 
finite  amount  of  Ught.  Hence  the  conclusion  to  be  adopted 
is  that  the  sun  is  in  the  midst  of  an  aggregation  of  at  least 
several  hundred  millions  of  stars  which  form  a  sort  of  system, 
and  that  beyond  and  far  distant  from  this  system  there  may 
be  other  somewhat  similar  systems  in  great  numbers,  which 
may  be  units  in  larger  systems,  and  so  on  without  Umit. 

It  is  conceivable  that  the  ether  is  not  infinitely  extensive, 
but  that  it  surrounds  the  stars  of  the  sidereal  system  (and 
other  stellar  systems  if  there  are  such)  as  the  atmospheres 
surround  the  planets.  Light  could  not  come  to  us  from 
beyond  its  borders,  however  many  sta,rs  might  exist  there, 
as  sound  caimot  come  to  the  earth  from  other  bodies  beyond 


470    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  269 

the  limits  of  its  atmosphere.  It  must  be  understood  that  this 
is  merely  a  suggestion  entirely  without  any  observational  basis. 

270.  The  Apparent  Distribution  of  the  Stars. —  The 
brighter  stars  are  quite  irregularly  distributed  over  the  sky, 
but  a  careful  examination  of  the  fainter  of  even  those  which 
can  be  seen  with  the  unaided  eye  shows  that  they  are  con- 
siderably more  numerous  in  and  near  the  Milky  Way  than 
elsewhere.  When  those  stars  which  can  be  seen  only  with 
the  help  of  a  telescope  are  included,  the  condensation  toward 
the  Milky  Way  is  still  more  pronounced. 

Precise  numbers  for  all  the  stars  are  known  only  to  the 
ninth  magnitude ;  but  the  star  counts  of  the  Herschels,  and 
especially  the  work  of  Chapman  and  Melotte,  go  much 
further  and  give  what  are  very  probably  approximately 
correct  results  down  to  the  seventeenth  magnitude.  Since 
the  stars  are  apparently  condensed  toward  the  Milky  Way, 
it  is  natural  to  use  its  plane  as  the  fundamental  plane  of 
reference.  According  to  E.  C.  Pickering  the  north  pole  of 
the  Galaxy  is  in  right  ascension  190°  and  its  decHnation  is 
+  28°.  The  Milky  Way  is  very  irregular  in  outline,  and  it 
is  difficult  to  locate  its  center ;  but  its  median  fine  is  possibly 
not  quite  a  great  circle,  from  which  it  follows  that  the  sun 
is  somewhat  out  of  the  plane  near  which  the  stars  are  con- 
gregated. 

Let  the  center  of  the  Milky  Way  be  the  circle  from  which 
galactic  latitudes  are  counted.  Chapman  and  Melotte 
divided  the  sky  up  into  eight  zones,  the  first  including  the 
belt  of  galactic  latitude  0°  to  ±  10°,  the  second  the  two  belts 
from  ±  10°  to  ±  20°,  the  third  the  two  belts  from  ±  20°  to 
±  30°,  the  fourth  from  ±  30°  to  ±  40°,  the  fifth  from  ±  40° 
to  ±  50°,  the  sixth  from  ±  50°  to  ±  60°,  the  seventh  from 
±  60°  to  ±  70°,  and  the  eighth  the  regions  from  ±  70°  to 
±  90°  around  the  galactic  poles.  With  the  belts  numbered 
in  this  order  they  found  for  the  average  number  of  stars  in 
each  magnitude  ip.  10  square  degrees  the  results  given  in 
Table  XV. 


CH.  xni,  270]        THE    SIDEREAL  UNIVERSE 


471 


Table  XV 


Zone 

I 

II 

III 

IV 

V 

VI 

VII 

VIII 

Galactic 
Latitude 

Oto±10° 

±  10°  to 
±20° 

±  20°  to 
±30° 

±30°  to 
±40° 

±40°  to 
±60° 

±50°  to 
±60° 

±  60°  to 
±70° 

±70°  to 
±90° 

Mag. 

1  to  5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 
17 

0.27 
0.7 
2.6 
8.0 
24 
62 
157 
363 
798 
1,642 
3,253 
6,150 
11,540 

0.23 
0.7 
2.3 
7.0 
21 
55 
135 
311 
658 
1,354 
2,650 
4,936 
9,170 

0.15 
0.5 
1.8 
6.1 
18 
50 
123 
280 
569 
1,142 
2,080 
3,680 
6,350 

0.11 
0.4 
1.5 
4.8 
14 
38 
93 
199 
409 
770 
1,390 
2,340 
3,980 

0.11 
0.3 
1.2 
3.8 
10 
28 
63 
136 
276 
531 
940 
1,680 
2,870 

0.11 
0.3 
1.1 
3.4 
10 
26 
62 
141 
295 
572 
1,050 
1,830 
3,100 

0.13 
0.3 
1.1 
3.2 
9 
22 
52 
115 
240 
482 
916 
1,630 
2,990 

0.13 
0.3 
1.1 
3.1 
8 
20 
47 
100 
205 
392 
773 
1,400 
2,610 

Total 

24,000 

19,300 

14,300 

9,240 

6,540 

7,090 

6,460 

5,560 

Three  things  follow  from  this  table:  (a)  Stars  of  all 
magnitudes  down  to  the  seventeenth  are  more  numerous  in 
the  plane  of  the  Milky  Way  than  near  its  poles.  Since  the 
only  reasonable  supposition  is  that  the  nearer  stars  are  dis- 
tributed more  or  less  uniformly  with  no  special  relations  to 
the  Milky  Way,  it  follows  from  the  fact  the  bright  stars 
are  condensed  near  the  Milky  Way  that  some  of  them  are 
very  distant.  That  is,  the  stars  differ  greatly  in  absolute 
luminosity,  a  conclusion  confirmed  by  direct  evidence. 
(6)  The  decrease  in  the  number  of  stars  is  on  the  average 
gradual  from  the  Milky  Way  to  its  poles,  showing  that  the 
sun  is  actually  in  the  midst  of  the  clouds  of  stars  on  which  the 
table  is  based,  (c)  The  relative  condensation  in  the  plane 
of  the  Milky  Way  is  greater,  the  fainter  the  stars.  This 
proves  that  the  stars  are  not  only  much  more  numerous 
near  the  plane  of  the  Milky  Way,  but  also  that  they  extend 
to  much  greater  distances  in  this  plane  than  in  the  direction 


472    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  Z/O 


Fig.  166.  —  Great  star  clouds  in  Sagittarius.       Photographed  by  Barnard  at 
the  Yerkes  Observatory. 


CH.  XIII,  271]        THE    SIDEREAL  UNIVERSE  473 

of  its  poles.  The  counts  of  stars  by  Kapteyn,  based  in  part 
on  the  visual  observations  of  Sir  John  Herschel,  give  still 
greater  relative  condensation  in  the  plane  of  the  Milky  Way, 
and  still  more  strongly  confirm  this  conclusion. 

271.  The  Form  and  Structure  of  the  Milky  Way.  —  Before 
atteinpting  to  arrive  at  a  more  precise  conclusion  regarding 
the  distribution  of  the  stars  in  space,  it  is  desirable  to  obtain 
a  better  idea  of  the  form  and  properties  of  the  Milky  Way. 

As  has  been  stated,  the  center  of  the  Milky  Way  is  nearly 
a  great  circle  around  the  celestial-  sphere.  Its  greatest 
northerly  declination  (45°  to  65°)  is  at  right  ascension  zero 
in  the  constellation  Cassiopeia,  where  it  is  about  20°  wide. 
It  extends  from ;  this  point  southeastward  across  Perseus 
with  very  irregular  outlines  (Map  1,  Art.  82),  and  narrows 
down  where  it  crosses  the  borders  of  Taurus  to  a  width 
of  about  5°.  It  then  bulges  wider  in  Monoeeros  and  across 
the  northeast  comer  of  Canis  Major.  Farther  sOuth  in 
Argo,  with  its  several  divisions,  it  becomes  as  much  as  30° 
wide,  but  its  borders  are  irregular,  it  is  broken  throi^gh  by 
vacant  lanes,  one  of  which  in  its  center  is  called  the  "  coal 
sack,"  and  at  right  ascension  about  9  hours  and  declination 
45°  south  a  dark  gap  stretches  almost  across  it.  After 
reaching  its  most  southerly  point  in  Crux  it  stretches  out  in 
irregular  outline  through  Centaurus,  part  of  Musca,  Circinus, 
Norma,  and  then  north  again  into  Ara,  Lupus,  and  Scorpius. 
In  Scorpius  and  in  Sagittarius  to  the  east  are  some  of  the  most 
remarkable  star  clouds  in  the  heavens.  Fig.  166.  Barnard's 
photographs  of  these  regions  show  countless  suns  massed 
in  banks,  with  intervening  dark  lanes,  the  whole  often 
enveloped  by  a  soft  nebulous  haze  (see  Fig.  167).  North- 
east of  Scorpius  he  Ophiuchus,  Serpens,  and  Aquila.  From 
Aquila  and  Ophiuchus  northward  through  Vulpecula  and 
Cygnus  to  Cepheus,  the  Milky  Way  is  divided  longitudinally 
by  a  rift  of  varying  width  and  form.  This  bifurcation,  which 
extends  through  more  than  50°  of  its  length,  is  one  of  its 
most  remarkable  features.     In  Cepheus  the  two  branches 


474    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xin,  271 

join  and  reach  on  into  Cassiopeia,  where  the  description  of 
the  Milky  Way  began. 

It  is  obvious  that  the  stars  do  not  form  any  simple  system. 
It  seems  probable  that  the  Galaxy  is  composed  of  a  large 


Fig.   167.  —  The  region  of  Rho  Ophiuehi.     Photographed  by  Barnard. 

number  of  star  clouds,  each  with  pecuUarities  of  its  own, 
but  having  relations  to  the  whole  mass  of  stars.  Since  the 
Milky  Way  is  roughly  in  the  form  of  a  great  discus,  or 
"  grindstone  "  as  Herschel  called  it,  the  prevailing  motions 
must  be  in  its  plane  in  order  to  have  preserved  its  shape. 


CH.  XIII,  271]        THE    SIDEREAL  UNIVERSE      .  475 

This  does  not  mean  that  the  relative  velocities  would  need  to 
be  great  enough  to  be  easily  observed ;  they  would,  in  fact, 
be  very  shght  as  seen  from  the  enormous  distances  separating 
the  stars  from  the  earth. 

XXII.   QUESTIONS 

1.  Prove  that  the  magnitudes  of  stars  of  equal  absolute  bright- 
ness are  proportional  to  the  squares  of  their  distances. 

2.  Prove  that,  under  the  hypothesis  of  the  second  paragraph 
of  Art.  269,  the  ratio  of  the  number  of  stars  of  any  magnitude 
to  the  number  of  the  next  magnitude  brighter  is  3.98. 

3.  If  there  are  2000  stars  of  magnitude  6  to  6,  and  if  the  ratio 
for  successive  magnitudes  were  3.98,  how  many  stars  would  there 
be  of  magnitude  16  to  17? 

4.  Prove  that  the  effectiveness  of  a  given  mass  in  screening 
off  light  is  inversely  proportional  to  the  radius  of  the  particles  into 
which  it  is  divided. 

5.  Show  in  detail  how  it  follows  from  Table  XV  and  the  as- 
sumption under  (a)  that  some  of  the  bright  stars  are  very  distant. 
How  many  of  the  20  first-magnitude  stars  have  parallaxes  greater 
than  0".2  (see  Table  XVI)  ? 

6.  At  what  distance,  expressed  in  parsecs  (Art.  272),  would  the 
sun  be  a  flrst-magnitude  star  ?  A  sixth-magnitude  ^tar  ?  If  Canopus 
has  a  parallax  of  0".006,  how  does  its  absolute  brightness  compare 
with  that  of  the  sun? 

7.  Prove  that  the  area  of  one  hemisphere  of  the  sky  is  92,000 
times  the  apparent  area  of  the  sun. 

8.  Prove  in  detail  that  conclusion  (b)  of  Art.  270  follows  from 
Table  XV. 

9.  At  what  time  of  the  year  does  the  portion  of  the  Milky  Way 
which  is  divided  by  a  longitudinal  rift  pass  the  meridian  at  8  p.m.  ? 
If  possible,  observe  it. 

10.  Draw  a  diagram  and  show  that  the  fact  that  the  central 
line  of  the  Milky  Way  is  not  quite  a  great  circle  proves  that  the 
solar  system  is  not  in  the  center  of  the  disk  of  stars  of  which  the 
Milky  Way  is  composed. 

11.  The  fact  that  the  Milky  Way  is  very  oblate  implies  that  it 
has  large  moment  of  momentum  about  an  axis  perpendicular  to 
its  plane.  What  inference  do  you  draw  respecting  the  general 
niotions  of  stars  in  exactly  opposite  parts  of  the  Milky  Way  ? 

12.  If  all  visible  objects  belong  to  the  Galaxy,  is  it  possible  to 
prove  the  rotation  of  the  Milky  Way  by  observations  of  the  stars? 


476     AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  271 

13.  What  observational  evidence  disproves  the  hypothesis  that 
there  are  infinitely  many  galaxies  distributed  with  approximate 
uniformity,  but  separated  from  one  another  by  distances  which 
are  enormous  compared  to  their  dimensions? 


II.  Distances  and  Motions  of  the  Staks 

272.  Direct  Parallaxes  of  the  Nearest  Stars.  —  One  of 

the  proofs  that  the  earth  revolves  around  the  sun  is  that  the 
apparent  directions  of  the  nearest  stars  vary  with  the  posi- 
tion of  the  earth  in  its  orbit  (Art.  51).  •  The  difference  in 
direction  of  a  star  as  seen  from  two  points  separated  from 
each  other  by  the  mean  distance  from  the  earth  to  the  sun 
is  the  parallax  of  the  star ;  or,  in  other  terms,  the  parallax 
of  the  star  is  the  angle  subtended  by  the  mean  radius  of  the 
earth's  orbit  as  seen  from  the  star  (Fig.  35).  If  the  parallax  , 
were  one  second  of  arc,  the  distance  of  the  star  would  be 
206,265  times  ^  the  mean  distance  from  the  earth  to  the  sun. 
This  distance,  which  is  a  very  convenient  unit  in  discussing 
the  distances  of  the  stars,  is  called  the  parsec,  and  for  most 
practical  purposes  it  may  be  taken  equal  to  200,000  astro- 
nomical units,  or  20,000,000,000,000  miles.  It  is  the  distance 
that  Ught  travels  in  about  3.3  years. 

The  stars  are  so  remote  that  the  problem  of  measuring  their 
parallaxes  is  one  of  great  practical  difficulty.  Alpha  Cen- 
tauri,  the  nearest  known  star,  has  a  parallax  of  only  0".75. 
That  is,  its  difference  in  direction  as  seen  from  two  points  on 
the  earth's  orbit,  separated  by  the  distance  from  the  earth 
to  the  sun,  is  the  same  as  the  difference  in  direction  of  an 
object  at  the  distance  of  10.8  miles  when  seen  first  with  one 
eye  and  then  with  the  other.  Not  only  is  the  difference  in 
the  apparent  position  of  a  star  very  small  as  seen  from  dif- 
ferent parts  of  the  earth's  orbit,  but  it  can  be  determined 
only  from  observations  separated  by  a  number  of  months 

'  This  number  is  the  number  of  seconds  in  the  arc  of  a  circle  which  equals 
its  radius  in  length. 


CH.  XIII,  272]        THE    SIDEREAL  UNIVERSE  477 

during  whieh  climatic  conditions  and  the  instruments  may 
have  appreciably  changed. 

The  best  direct  means  of  determining  the  parallax  of  a 
star  is  by  comparing,  at  various  times  of  the  year,  its  appar- 
ent position  with  the  positions  of  more  distant  stars.  Let  S, 
Fig.  168,  represent  a  star  whose  parallax  is  required,  and  S' 
a  much  more  distant  star.  When  the  earth  is  at  Ei  the 
angular  distance  between  them  is  ^  SEiS' ;  when  the  earth 
is  at  E2,  it  is  SE^S'.  The  parallax  of  /S  is  Z  EiSE^;  the 
parallax  of  S'  is  EiS'E^,  which  will  be  negligible  if  S'  is  suffi- 
ciently remote.  It  easily  follows  from  the  geometry  of  the 
figure  that  the  parallax  of  S  mimis  the  parallax  of  S'  equals 
the  difference  of  the  measured  angles  SEiS'  and  SEaS'. 


Fia.  168.  —  Determination  of  parallax  from  apparent  changes  in  relative 
positions  of  stars. 

Hence,  if  the  parallax  of  »S'  is  inappreciable,  the  parallax  of 
S  can  be  found. 

In  practice  the  position  of  Sis  measured  with  respect  to 
a  number  of  comparison  stars.  At  present  the  work  is  done 
almost  entirely  by  photography.  Plates  of  a  star  and  the 
surrounding  region  are  secured  at  different  times  of  the  year, 
and  the  distances  between  the  stars  are  measured  under  a 
microscope  on  a  machine  designed  for  the  purpose.  The 
scale  of  the  photograph  is  proportional  to  the  focal  length 
of  the  telescope,  and  consequently  for  this  purpose  only 
large  and  excellent  instruments  are  of  value. 

With  present  means  of  measurement,  a  parallax  of  0".02 
or  less  cannot  be  determined  with  sufficient  accuracy  to  be 
of  much  value;  in  fact,  the  probable  error  in  one  of  0".05 
is  large.    The  great  distances  of  the  stars  can  be  inferred 


478    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  272 

from  the  fact  that  only  about  100  are  known  whose  parallaxes 
come  within  the  wider  of  these  Umits. 

The  distances  of  stars  whose  parallaxes  are  0".2  or  greater 
can  be  measured  with  an  error  not  exceeding  about  25  per 
cent  of  the  quantity  to  be  determined.  There  are  at  present 
19  such  stars  known,  9  of  which  are  too  faint  to  be  seen  with- 
out optical  aid.  These  stars  are  given  in  Table  XVI.  When 
the  distance  of  a  star  of  known  magnitude  has  been  deter- 
mined, the  total  amount  of  Hght  it  radiates,  or  its  luminosity, 
as  compared  with  the  sun  can  be  computed.  The  luminosity 
of  each  of  the  nineteen  stars  is  given  in  the  fifth  column. 


Table  XVI 


Star 

Mag- 

Paral- 

Distance 

Luminosity 

Mass 

Velocity 

nitude 

lax 

(Parsecs) 

(SnN  =  l) 

CSUN=1) 

(Mi.  per  Sec.) 

a  Centauri     . 

0.3 

0.76 

1.32 

2.0 

1.9 

20 

Lalande  21,185  . 

7.6 

0.40 

2.50 

0.009 

35+ 

Sirius   . 

-1.6 

0.38 

2.63 

48.0 

3 

4 

11 

T  Ceti  .     . 

3.6 

0.33 

3.00 

0.50 

20 

Procyon    .     .     . 

0.5 

0.32 

3.13 

9.7 

1 

3 

12 

C.  Z.  5"  243 

8.3 

0.32 

3.13 

0.007 

170 

e  Eridani  . 

3.3 

0.31 

3.23 

0.79 

14 

61  Cygni  .     .     . 

5.6 

0.31 

3.23 

0.10 

63 

Lacaille  9352 

7.4 

0.29 

3.45 

0.019 

72 

Pos.  Med.  2164  . 

8.8 

0.29 

3.45 

0.006 

23+ 

e  Indi   .      .     .      . 

•4.7 

0.28 

3.57 

0.25 

54 

Groombridge  34 

8.2 

0.28 

3.57 

0.010 

30+ 

OA(N.)  17,415   . 

9.3 

0.27 

3.70 

0.004 

14+ 

Krueger  60    .     . 

9.2 

0.26 

3.85 

0.005 

11+ 

Altair        .     .     . 

0.9 

0.24 

4.17 

12.3 

22 

Tj  Cassiopeise* 

3.6 

0.20 

5.00 

1.4 

1 

0 

20 

ff  Draconis 

4.8 

0.20 

5.00 

0.5 

30 

Lalande  21,258  . 

8.9 

0.20 

5.00 

0.011 

66+ 

OA(N.)  11,677  . 

9.2 

0.20 

5.00 

0.008 

45+ 

The  19  stars  of  Table  XVI  together  with  our  sun  occupy  a ' 
sphere  whose  radius  is  5  parsecs.     If  they  were  uniformly 
distributed  in   this   space,  the  distance  between   adjacent 
stars  would  be  about  3.7  parsecs,  or  12.2  light  years.     In 
view  of  the  fact  that  a  number  of  stars  in  the  list  are  far 


CH.  XIII,  272]        THE    SIDEREAL  UNIVERSE  -       479 

below  the  limits  of  visibility  without  optical  aid,  it  is  reason- 
able to  suppose  that  there  may  be  a  considerable  number  of 
others  within  5  parsecs  of  the  sun  which  are  as  yet  undis- 
covered. 

It  should  not  be  supposed  that  attempts  have  been  made 
to  measure  the  parallaxes  of  all  stars  brighter  than  the 
ninth,  or  even  the  sixth,  magnitude.  The  process  is  exces- 
sively laborious,  and  only  those  stars  are  selected  which  are 
beheved  to  be  within  measurable  distance,  or  which  are 
objects  of  especial  interest  for  other  reasons.  A  star  with  a 
given  motion  across  the  line  of  sight  will  apparently  move 
faster  the  nearer  it  is  to  the  observer.  Consequently, 
those  stars  will  be  nearest  on  the  average  whose  -proper 
motions,  as  they  are  called,  are  greatest.  As  a  rule  only  those 
stars  are  examined  for  parallax  which  have  been  found  to 
have  large  proper  motions. 

Under  the  hypotheses  that  the  stars  are  uniformly  dis- 
tributed throughout  the  space  occupied  by  the  Galaxy  and 
that  their  density  is  the  same  as  it  is  in  the  vicinity  of  the 
sun,  the  extent  of  the  stellar  universe  can  be  computed. 
Suppose  the  space  occupied  by  the  stars  is  spherical  in  shape 
and  that  there  are  500,000,000  of  them.  Then  it  turns  out 
that,  under  the  hypotheses  adopted,  the  radius  of  this  sphere 
is  1500  parsecs,  or  5000  light-years.  Since  the  Galaxy  is  very 
much  flattened,  the  distance  to  its  poles  is  probably  only  a 
few  hundred  parsecs  while  the  borders  of  its  periphery  are 
probably  several  thousand  parsecs  from  its  center. 

One  very  interesting  and  important  conclusion  follows  from 
Table  XVT,  and  that  is  that  the  luminosities  of  the  stars 
vary  enormously.  For  example,  Sirius  radiates  12,000 
times  as  much  Ught  as  OA(N.)  17,415.  These  differences  in 
luminosity  may  be  due  to  the  fact  that  some  stars  are  larger 
than  others,  or  at  least  partly  to  the  fact  that  some  are 
intrinsically  more  brilliant  than  others.  Probably  both 
factors  are  important.  Some  stars  are  certainly  much  more 
massive  than  others,  and  the  table  gives  examples  of  stars 


480    AN   INTRODUCTION   TO  ASTRONOMY  [ch.  xiii,  272 

whose  masses  differ  very  much  less  than  their  luminosities. 
For  example,  while  the  mass  of  Sirius  is  only  3.4  times  that 
of  the  sun,  its  luminosity  is  48  times  as  great.  But  Sirius  is  a 
double  star  and  presents  in  its  own  system  a  still  more 
remarkable  contrast.  The  mass  of  the  brighter  component 
is  approximately  twice  that  of  the  fainter  one,  but  in  lumi- 
nosity it  is  at  least  5000  times  greater.  There  are  other  stars, 
such  as  Rigel  and  Canopus,  which,  though  they  are  so  remote 
that  no  evidence  of  their  having  measurable  parallaxes  has 
,  been  found,  shine  with  the  greatest  brilliancy.  Their  lumi- 
nosity must  be  at  least  several  thousand  times  that  of  the  sun. 
In  fact,  the  average  luminosity  of  the  stars  visible  to  the 
unaided  eye  probably  exceeds  that  of  the  sun  several  hun- 
dred fold.  It  must  not  be  assumed  from  this  that  the 
luminosity  of  the  sun  is  below  the  average,  for  it  is  exceeded 
in  luminosity  by  only  five  of  the  19  stars  in  the  table. 

In  order  to  determine  the  velocity  of  a  star  its  motion  both 
along  and  across  the  line  of  sight  must  be  found.  The  proper 
motions  of  all  the  stars  in  Table  XVI  are  known,  but  the 
radial  velocities  of  six  of  them  are  unknown ;  in  these  cases 
a  plus  sign  is  placed  after  the  number  giving  the  velocity 
because  the  radial  component  is  not  known.  It  follows  from 
the  table  that  the  less  luminous  stars  move  with  much 
higher  velocities  than  the  brighter  ones.  The  average  speed 
of  those  five  stars  whose  luminosities  exceed  the  sun  is  17 
miles  per  second,  while  the  average  speed  of  the  six  whose 
luminosities  are  less  than  0.01  that  of  the  sun  is  more  than 
50  miles  per  second.  Since  the  more  luminous  stars  are 
almost  certainly  the  more  massive,  it  follows  that  the  more 
massive  stars  move  more  slowly  than  the  smaller  ones. 

One  may  inquire  to  what  extent  reliance  can  be  put  in 
conclusions  based  on  only  19  stars.  When  compared  to 
hundreds  of  millions  the  number  is  ridiculously  small,  but  all 
the  conclusions  which  have  been  stated  are  strongly  supported 
by  the  evidence  furnished  by  the  much  more  numerous  stars 
having  smaller  and  less  accurately  determined  parallaxes. 


CH.  xm,  273]        THE    SIDEREAL  UNIVERSE  481 

273.  Distances  of  the  Stars  from  Proper  Motions  and 
Radial  Velocities.  —  The  parallaxes  of  possibly  100  stars  have 
been  determined  by  direct  means  with  considerable  accuracy. 
Probably  not  over  1000  are  within  reach  of  present  instru- 
ments and  methods.  Are  astronomers  doomed  to  remain 
in  ignorance  as  to  the  distances  of  all  the  other  stars  which 
fill  the  sky?  By  no  means.  There  are  several  indirect 
methods  of  finding  the  average  distances  of  classes  of  stars. 

Consider  all  the  stars  of  a  large  class,  say  the  stars  of  the 
sixth  magnitude.  Suppose  they  are  moving  at  random; 
that  is,  that  they  do  not  tend  to  move  in  any  particular 
direction,  or  with  any  particular  speed.  Suppose  both  their 
proper  motions  and  their  radial  velocities  have  been  deter- 
mined by  observation.  Under  these  hypotheses  as  many 
stars  win  be  approaching  as  receding,  and  the  velocities  of 
approach  will  average  the  same  as  those  of  recession.  Also, 
the  proper  motions  will  be  as  numerous  and  as  large  in  one 
direction  as  in  the  opposite.  The  extent  to  which  these 
conditions  are  fulfilled  is  a  measure  of  the  accuracy  of  the 
assumptions. 

-Whatever  the  individual  motions  of  the  class  of  stars 
under  consideration,  they  will  have  an  average  speed  of  mo- 
tion which  may  be  represented  by  V.  The  average  compo- 
nent of  motion  toward  or  from  the  observer  will  be  |F,  as  can 
be  shown  by  a  mathematical  discussion.  This  is  the  aver- 
age radial  velocity  as  determined  by  the  spectroscope,  and 
is  therefore  known.  The  average  component  at  right  angles 
to  the  line  of  sight  is  found  by  a  mathematical  discussion 
to  be  0.7854  V.  This  quantity  is  therefore  also  known 
because  V  has  been  given' by  spectroscopic  observations. 

Now  consider  the  proper  motions.  They  are  expressed 
in  angle,  and  they  depend  upon  the  distances  of  the  stars 
and  the  speed  with  which  they  move  across  the  line  of  sight. 
Since  both  the  linear  speed  across  the  line  of  sight  and  the 
angular  velocity,  or  proper  motion,  have  been  found,  the 
distances  of  the  stars  can  be  computed. 
2i 


482     AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  -^16 

The  hypotheses  on  which  this  discussion  has  been  made 
are  not  exactly  fulfilled,  and  the  necessary  modifications  of 
the  proposed  method  must  now  be  considered. 

274.  Motion  of  the  Sun  with  Respect  to  the  Stars.  —  Since 
the  stars  are  in  motion,  it  is  reasonable  to  suppose  that 
the  sun  is  moving  among  them.  Such  was  found  to  be 
the  case  by  Sir  William  Herschel  more  than  a  century  ago. 
He  proved  by  observations  extending  over  many  years  that 
the  apparent  distances  between  the  stars  in  the  direc- 
tion of  the  constellation  Hercules  are  increasing,  on  the 
average,  and  that  they  are  decreasing  in  the  exactly  opposite 
part  of  the  sky.  He  interpreted  this  as  meaning  that 
the  sun  is  moving  toward  the  constellation  Hercules,  and 
it  is  obvious  that  this  would  explain  the  observed  phe- 
nomena; for,  as  objects  are  approached,  they  subtend 
larger  angles.  While  Herschel's  observations  gave  the 
direction  of  motion  of  the  sun,  they  did  not  give  its 
speed,  which  could  be  found  by  this  method  only  if  the 
distances  of  the  stars  were  known.  Since  the  distances  of 
only  a  few  stars  can  be  measured  directly,  there  is  little  hope 
of  determining  the  motion  of  the  sun  in  this  way  with  any 
considerable  degree  of  accuracy. 

The  spectroscope  has  been  used  to  determine  both  the 
direction  of  the  sun's  motion  and  also  the  rate  at  which  it 
moves.  Instead  of  finding  as  many  stars  approaching  as 
receding  in  every  part  of  the  sky,  as  was  assumed  in  the  dis- 
cussion in  Art.  273,  it  has  been  found  that  the  stars  in  the 
direction  of  the  constellation  Hercules  on  the  whole  are 
relatively  approaching  the  sun,  while  those  in  the  opposite 
direction  are  relatively  receding.  This  means  that  with 
respect  to  the  stars  which  were  observed  the  sun  is  moving 
toward  Hercules. 

The  best  determination  of  the  direction  of  the  sun's  motion 
from  proper  motions  of  the  stars  is  by  Lewis  Boss,  who  based 
his  discussion  on  the  6188  stars  in  his  catalogue.  The  best 
spectroscopic  determination  is  by  W.  W.  Campbell,  who 


CH.  XIII,  274]        THE    SIDEREAL  UNIVERSE  483 

based  his  discussion  on  the  radial  velocities  of  1193  stars 
measured  at  the  Lick  Observatory  and  its  branch  in  South 
America.     The  results  of  these  determinations  are  as  follows : 


Right 
Ascension 

Declination 

Speed 

Solar  Apex  (Boss)     .     . 
Solar  Apex  (Campbell) . 

270°.5  ±  1°.5 
268°.5  ±  2°.0 

+  34°.3  ±  1°.3 
+  25°.3±  1°.8 

? 
12  mi.  per  sec. 

The  agreement  of  these  results  in  right  ascension  is  remark- 
able, and  the  disagreement  in  declination  is  small  consider- 
ing the  difference  in  the  methods  and  the  stars  used. 

The  number  of  stars  used  by  Boss  in  his  determination  of 
the  direction  of  the  motion  of  the  sun  is  so  great  that  he  could 
divide  them  up  into  separate  groups  and  make  the  discussion 
for  each  one  separately.  He  took  the  stars  of  various  galac- 
tic latitudes  and  obtained  essentially  the  same  result  for 
each  group.  Dyson  and  Thackeray  found  from  another  (the 
Groombridge)  Ust  of  3707  stars  that  the  declination  of  the 
apex  of  the  sun's  way  increases  from  +16°  for  the  brightest 
stars  to  +  43°  for  those  from  magnitude  8.0  to  8.9.  This 
was  confirmed  by  Comstock,  who  found  even  a  greater  decli- 
nation for  the  apex  of  the  sun's  way  as  determined  from  still 
fainter  stars,  but  the  result  must  be  accepted  with  reserve 
until  it  is  confirmed  by  a  discussion  depending  on  a  much 
larger  and  better  distributed  list  of  stars.  The  spectra  of  the 
stars  are  divided  into  a  number  of  classes  (Art.  295),  and  it 
was  found  both  by  Boss  and  by  Dyson  and  Thackeray  that 
the  dechnation  of  the  apex  of  the  sun's  way  is  about  12° 
greater  when  determined  from  stars  of  Secchi's  second  type 
than  it  is  when  determined  from  stars  of  the  first  type.  But 
the  results  altogether  indicate  that  the  sun  is  moving,  rela- 
tively to  the  few  thousand  brightest  stars,  toward  a  point 
whose  right  ascension  is  about  270°  and  whose  dechnation  is 
about  34°,  and  that  the  speed  of  relative  motion  is  about  12 
miles  per  second. 


484    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  274 

The  motion  of  the  sun  with  respect  to  the  stars  evidently 
reqiiires  some  modification  of  the  process  described  in  Art. 
273.  There  is,  however,  no  real  difficulty,  because  the  effect 
of  the  sun's  motion  can  be  avoided  by  considering  only  those 
components  of  the  proper  motions  of  the  stars  which  are  at 
right  angles  to  the  line  of  the  sun's  way. 

Campbell  made  a  determination  of  the  mean  parallaxes  of 
the  stars  down  to  magnitude  5.5  by  the  method  of  this 
article.  The  brighter  stars  were  not  sufficiently  ntmierous  to 
give  very  rehable  results.  He  found  that  the  mean  parallax 
of  stars  of  magnitudes  4.51  to  5.50  is  0".0125,  corresponding 
to  a  distance  of  80  parsecs.  This  volimie  is  4096  times 
that  occupied  by  the  20  nearest  stars,  and  if  the  stars  were 
uniformly  distributed  throughout  it,  the  total  number  of 
them  down  to  magnitude  5.50  would  be  81,920,  which  is 
much  in  excess  of  the  number  actually  observed. 

275.  Distances  of  the  Stars  from  the  Motion  of  the  Sun.  — 
The  parallaxes  of  only  a  comparatively  small  number  of  stars 
can  be  measured  directly  because  their  distances  are  so  enor- 
mously great  compared  to  the  diameter  of  the  earth's  orbit. 
If  the  orbit  of  the  earth  were  as  large  as  that  of  Neptune,  the 
problem  would  be  much  easier  because  of  the  larger  base  line 
which  could  be  used.  But  the  sun's  motion  can  be  made  to 
afford  an  indefinitely  large  base  line  in  statistical  discussions, 
as  will  now  be  shown.    . 

Suppose  first  that  all  of  the  stars  of  the  observable  sidereal 
universe  except  the  sun  are  relatively  at  rest.  The  motion 
of  the  sun  among  them  will  give  them  an  apparent  displace- 
ment, or  proper  motion,  in  the  direction  opposite  to  that  in 
which  it  is  moving.  The  farther  a  star  is  away  the  smaller 
this  proper  motion  will  be.  If  a  star  is  so  far  away  that  no 
displacement  due  to  the  sun's  motion  can  be  observed  in  one  * 
year,  then  10  years,  100  years,  or  any  other  necessary  num- 
ber of  years  may  be  used.  Eventually  the  effect  of  the  sun's 
motion  will  be  observable.  Since  the  sun  travels  about  4 
astronomical  units  per  year,  it  follows  that  the  parallax  of  a 


CH.  XIII,  275]         THE  SIDEREAL  UNIVERSE  485 

star  is  one  fourth  of  that  part  of  its  annual  proper  motion 
which  is  due  to  the  motion  of  the  sun. 

The  false  hypothesis  that  all  the  stars  except  the  sun  are 
relatively  at  rest  has  greatly  simjpUfied  the  problem.  As  a 
matter  of  fact,  the  stars  are  moving  with  respect  to  one  an- 
other in  various  directions  and  with  various  speeds,  and  the 
proper  motion  of  a  star  is  due  both  to  its  own  motion  and  also 
to  the  motion  of  the  sun  with  respect  to  the  system.  Since 
the  actual  motion  of  any  particular  star  is  in  general  un- 
known, it  is  necessary  to  take  the  average  motions  of  many, 
and  then  the  results  will  be  consistent,  for  the  motion  of  the 
sun  is  defined  with  respect  to  the  many.  For  any  class  of 
stars  the  average  proper  motion  perpendicular  to  the  direc- 
tion of  the  sun's  motion  will  be  zero,  while  the  average  proper 
motion  in  the  direction  of  the  sun's' motion  will  depend  only 
on  their  distance  and  the  speed  of  the  sun. 

This  statistical  study  of  the  stars  was  taken  up  about  20 
years  ago  by  Kapteyn,  of  Groningen,  who  pursued  it  with 
rare  skill  and  great  industry.  A  number  of  other  astronomers 
have  also  made  important  contributions  to  the  subject.  It 
is  interesting  to  note  the  different,  kinds  of  work  which  -con- 
tribute to  the  final  results.  In  the  first  place,  the  proper 
motions  of  the  stars  are  involved.  They  are  obtained  from 
two  or  more  determinations  of  apparent  position  separated 
by  considerable  intervals.  In  fact,  the  longer  the  intervals 
the  more  accurately  are  the  proper  motions  determined.  In 
the  second  place,  the  spectroscope  is  of  fundamental  impor- 
tance because  it  furnishes  the  motion  of  the  sun  with  respect 
to  the  stars.  Since  certa'n  classes  of  stars  may  be  moving  as 
a  whole  with  respect  to  other  classes  (Art.  278),  it  follows  that 
the  spectroscopic  determination  of  the  motion  of  the  sun 
should  depend  upon  all  those  stars  whose  distances  are 
sought  from  their  proper  motions.  At  present  the  radial 
velocities  of  stars  fainter  than  the  sixth  magnitude  can  be 
obtained  only  by  costly  long  exposures,  and  the  practical 
limits  do  not  reach  beyond  the  eighth  magnitude.    On  the 


486    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  275 

other  hand,  the  determination  of  the  proper  motions  of  stars 
many  magnitudes  fainter  offers  no  observational  difficulties. 

276.  Kapteyn's  Results  Regarding  the  Distances  of  the 
Stars.  —As  will  be  seen  in  Art.  295,  most  of  the  stars  are  of 
two  principal  spectral  types.  Type  I,  of  which  Sirius  and 
Vega  are  conspicuous  examples,  are  white  or  bluish  white. 
Their  spectra  are  characterized  by  absorption  lines  due  to 
hydrogen  in  their  atmospheres.  They  are  intensely  hot  and 
probably  always  of  large  mass.  Type  II  are  the  yellowish 
stars,  of  which  the  sun,  Capella,  and  Arcturus  are  examples. 
The  atmospheres  of  these  stars  contain  many  metals. 

Kapteyn  derived  formulae  giving  the  mean  parallaxes  of 
all  stars' of  each  magnitude,  and  also  the  mean  distances  of 
stars  of  each  spectral  type  separately.  Table  XVII  gives 
Kapteyn's  results  transformed  from  parallax  to  parsecs  and, 
using  Campbell's  more  recent  determination  of  the  rate  of 
motion  of  the  sun. 

Table  XVII 

Distances  in  Parsecs' 


Magnitude 

All  Stabs 

Spectral 
Type  I 

Spectral 
Type  II 

1 

24.2 

39.4 

16.8 

2 

31.0 

"50.5 

21.6 

3 

39.7 

64.7 

27.6 

4 

50.9 

82.9 

35.4 

5 

65.3 

106.3 

45.4 

6 

83.7 

136.3 

58.2 

7 

107.3 

174.7 

74.7 

8 

137.5 

224.0 

95.7 

9 

176.3 

287.2 

122.7 

10 

226.1 

368.3 

157.4 

11 

289.8 

472.1 

201.7 

12 

371.6 

605.3 

258.6 

13 

476.4 

776.0 

331.6 

14 

610.8 

994.9 

425.2 

15 

783.0 

1275.5 

545.0 

'  One  parsec  equals  200,000  astronomical  units,  or  in   round  numbers 
20,000,000,000,000  miles. 


CH.  XIII,  277]        THE   SIDEREAL  tJNIVERSE  487 

It  must  be  remembered  that  Table  XVII  gives  mean  results 
derived  from  the  proper  motions  and  radial  velocities  of 
many  stars.  The  results  may  be  in  error  for  the  first  few 
magnitudes  because  there  are  not  enough  bright  stars  to 
make  the  statistical  method  reUable.  They  may  also  be  in 
error  for  the  fainter  stars  because  these  stars  were  not  used  in 
deriving  the  formulae  by  which  the  computations  were  made. 

If  the  table  is  correct,  the  sun  is  far  below  the  average  of 
the  stars  in  brilliancy.  According  to  the  measures  of  Wol- 
laston,  Bond,  and  Zollner  its  magnitude  on  the  stellar  basis 
is  -  26.7,  or  it  gives  us  120,000,000,000  times  as  much  light 
as  a  first-magnitude  star.  Since  the  light  received  from 
a  body  varies  inversely  as  the  square  of  its  distance,  at  the 
mean  distance  of  the  first-magnitude  stars  the  sun  would 
send  us  only  0.005  as  much  light  as  comes  from  a  first-magni- 
tude star.  That  is,  the  first-magnitude  stars  average  about 
200  times  as  brilliant  as  the  sun.  It  must  not  be  concluded 
from  this  that  the  stars  of  all  magnitudes  average  so  much 
more  brilliant  than  the  sun,  for  those  of  the  first  magnitude 
are  a  group  selected  because  of  their  great  brilliancy." 

277.  Distances  of  Moving  Groups  of  Stars.  —  If  the  two 
components  of  a  double  star  are  found  to  be  movjng  in  the 
same  direction  and  with  the  same  apparent  speed,  the  con- 
clusion to  be  drawn  is  that  they  are  relatively  close  together 
in  space  and  that  they  are  physically  connected ;  for,  if  they 
were  simply  in  the  same  direction  from  the  earth  without 
being  related,  their  apparent  motions  would  almost  certainly 
differ  either  in  speed  or  direction.  While  the  conclusion 
might  be  erroneous  in  the  case  of  only  two  stars,  it  could 
hardly  fail  to  be  true  if  many  stars  were  involved. 

The  study  of  the  proper  motions  of  the  stars  has  shown 
that  there  are  several  groups  which  have  sensibly  identical 
proper  motions;  or  rather,  as  the  result  of  perspective, 
there  are  many  stars  which  apparently  move  with  the  same 
speed  toward  a  common  point  in  the  sky.  These  groups 
are  widely  scattered  and  many  of  their  members  would  not 


488    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  277 

be  suspected  of  being  associated  with  the  oliiers  except  for 
the  equahty  of  their  motions.  For  example,  Sirius  belongs 
to  a  group  which  includes  five  of  the  stars  in  the  Big  Dipper. 
The  best-known  group  of  stars  of  the  type  under  considera- 
tion comprises  part  of  the  Hyades  cluster,  in  the  constellation 
Taurus,  and  some  neighboring  stars  scattered  over  an  area 
about  15°  in  diameter.  This  group,  which  includes  39  known 
stars,  was  exhaustively  discussed  by  Lewis  Boss.  The  stars, 
in  their  proper  motions,  all  seem  to  move  along  the  arcs 
of  great  circles.     Boss  found  that  the  great  circles  of  all 

the  stars  of  the 
Taurus  stream 
intersect  in  a 
common  point 
whose  right  as- 
cension and  dec- 
lination are,  for 
/  the  position  of  the 

'  equinox  in  1875, 

6  h.  7.2  m.  and  + 

6°  56'.    It  can  be 

/  shown  that    this 

0  ^  p  means    that    the 

Fig.  169.  —  Components  of  motion  in  moving  groups    stars  of  the  grOUp 
of  stars. 

are  movmg  in 
lines  parallel  to  the  line  from  the  observer  to  the  point  of 
intersection  of  the  circles.  That  is,  their  direction  of  motion 
is  defined  in  this  way,  and  since  the  stars  cover  a  consider- 
able area  in  the  sky  the  point  toward  which  they  are  moving 
is  very  well  determined. 

It  will  now  be  shown  that  if,  in  addition  to  the  data  already 
in  hand,  the  radial  velocity  of  one  of  the  stars  of  the  group 
can  be  obtained,  then  the  actual  motions,  the  distances,  and 
the  lummosities  of  all  of  them  can  be  determined.  Let  0, 
Pig.  169,  be  the  position  of  the  observer  and  OP  the  direction 
of  motion  of  the  stars  of  the  group.     Let  S  be  one  of  the 


CH.  XIII,  277]        THE   SIDEREAL  UNIVERSE  489 

stars  which  is  moving  in  the  known  direction  SA  with  an 
unknown  speed.  Suppose  the  component  SB  is  measured 
by  the  spectroscope.  Then,  since  the  angle  ASB,  which 
equals  the  angle  POS,  is  known,  the  whole  component  SA 
and  the  proper-motion  component  SC  can  be  computed. 
That  is,  the  actual  distance  SC  is  found  and  the  pl-oper 
motion  to  which  it  gives  rise  was  already  known.  There- 
fore the  distance  OS  can  be  computed.  Since  all  the  stars 
of  the  group  must  have  the  same  total  motion  SA,  for  other- 
wise they  would  not  remain  long  associated,  the  distances  of 
all  the  members  can  be  determined  from  their  respective 
proper  motions.  .  Of  course,  it  is  practically  advantageous  to 
measure  the  radial  velocities  of  many,  or  all,  of  the  members 
of  the  group.  When  the  distance  of  a  star  of  known  magni- 
tude has  been  found,  its  absolute  luminosity  can  be  com- 
puted. 

By  these  methods  Boss  found  that  the  Taurus  group  is  a 
globular  cluster  whose  center  is  distant  about  40  parsecs 
from  the  earth.  Since  its  apparent  diameter  is  about  15°, 
its  actual  diameter  is  about  10  parsecs.  There  is  a  slight  con- 
densation toward  the  center  of  the  cluster,  but  in  the  group 
as  a  whole  the  star  density  is  only  a  little  greater  than  it  is 
in  the  vicinity  of  the  sun.  The  distances  between  the  stars 
df  the  group  are  so  great  that  foreign  stars  could  pass  through 
it  without  having  their  motions  appreciably  disturbed.  In 
fact,  in  the  motion  of  the  cluster  it  certainly  sweeps  past 
other  stars  and  there  are  probably  several  strangers  now 
within  its  borders.  Boss  found  that  800,000  years  ago  the 
cluster  was  half  its  present  distance  and  its  apparent  size  was 
twice  that  at  present.  In  65,000,000  years  it  will  have 
receded  until  it  will  appear  from  the  earth  to  be  a  compact 
group  one  third  of  a  degree  in  diameter,  made  up  of  stars  of 
the  ninth  magnitude  and  fainter. 

All  the  39  stars  of  the  Taurus  cluster -are  much  greater  in 
light-giving  power  than  the  sun.  The  luminosities  of  even 
the  five  smallest  are  from  five  to  ten  times  that  of  the  sun, 


490    AN  INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  277 

while  tEe  largest  are  100  times  greater  in  light-giving  power 
than  our  own  luminary.  Their  masses  are  probably  much 
greater  than  that  of  the  sun. 

The  Ursa  Major  group  of  13  stars  is  another  wonderful 
system.  It  is  in  the  form  of  a  disk  whose  thickness  is  only 
4  or  5  parsecs  while  its  diameter  is  50  parsecs.  The  dis- 
tances of  the  members  of  this  group  from  the  sun  vary  from 
2.6  parsecs,  in  the  case  of  Sirius,  to  22  parsecs  for  the  stars  of 
the  Big  Dipper,  and  over  40  parsecs  in  the  case  of  Beta 
Aurigae.  The  luminosities  of  the  stars  vary  from  7  to  more 
than  400  times  that  of  the  sun. 

There  is  another  fairly  weU-estabhshed  group  in  Perseus 
which  was  discovered  almost  simultaneously  by  Kapteyn, 
Benjamin  Boss,  and  Eddington.  There  are  several  other 
probable  groups  in  which  the  proper  motions  are  so  small 
that  the  results  have  not  been  established  beyond  all  ques- 
tion. In  a  universe  of  many  stars  it  is  inevitable  that  there 
should  be  many  accidental  parallelisms  and  equalities  of 
motion.  Stars  are  at  present  regarded  as  forming  a  related 
group  only  if  there  is  something  quite  distinctive  about  their 
positions  or  motions. 

278.  Star-Streams.  —  In  1904  Kapteyn  announced  a  very 
important  discovery  respecting  the  motions  of  the  stars. 
He  found  that,  instead  of  moving  at  random,  most  of  the 
stars  belong  to  two  great  streams  having  well-defined  direc- 
tions of  motion.  Stars  in  all  parts  of  the  sky,  of  all  magni- 
tudes so  far  as  the  proper  motions  are  known,  and  of  all 
spectral  types,  partake  of  these  motions.  The  phenomena 
do  not  seem  to  be  local,  so  to  speak,  as  was  true  in  case  of 
the  groups  considered  in  Art.  277.  Yet  it  would  be  going 
too  far  to  conclude  that  all  the  stars  in  the  clouds  which  make 
up  the  Milky  Way  belong  to  these  streams,  for  the  discussion 
was  based  on  only  a  few  thousands  of  stars,  while  there  are 
hundreds  of  millions  in  the  sky.  It  seems  probable  that  the 
Galaxy  is  made  up  of  a  great  many  of  these  streams.  There 
is,  in  fact,  some  reason  to  believe  that  there  is  a  third  drift 


CH.  XIII,  279]        THE   SIDEREAL  UNIVERSE  491 

containing  stars  of  the  so-called  Orion  type.  But  the  evi- 
dence for  the  existence  of  the  two  streams  discovered  by 
Kapteyn  is  conclusive,  and  his  results  have  been  verified  by 
several  other  astronomers.  And  in  connection  with  the 
larger  problems  of  the  Milky  Way,  it  is  interesting  to  note 
that  both  streams  are  moving  parallel  to  its  plane. 

With  respect  to  the  sun  as  an  origin  the  points  toward 
which  the  stars  are  moving  are  : 

Apex  of  Drift  I :    Right  Ascension,  90° ; 
Declination,  —15°. 

Apex  of  Drift  II :    Right  Ascension,  288° ; 
Declination,  —64°. 

If  the  motion  of  the  sun  is  eliminated  and  the  stars  are 
considered  only  with  reference  to  one  another,  the  two 
streams  necessarily  move  in  opposite  directions.  With  this 
reference,  the  vertices  of  the  two  drifts  according  to  Edding- 
ton's  discussion  of  the  stars  in  Boss's  catalogue  are : 

Right  Ascensions,  94°,  274°; 

Declinations,  +12°,  -12°. 

About  60  per  cent  of  the  stars  on  which  the  discussion  was 
based  belong  to  Drift  I  and  40  per  cent  to  Drift  II.  They  are 
intermingled  in  space  so  that  one  set  of  stars  is  passing 
through  the  other.  Their  relative  velocity  is  about  24  miles 
per  second,  or  about  8  astronomical  units  per  year. 
■^  279.  On  the  Dynamics  of  the  Stellar  System.  —  The 
stars  are  at  least  several  hundred  millions  in  number,  they 
occupy  an  enormous  space,  and  they  are  moving  with  respect 
to  one  another  with  velocities  averaging  about  20  miles  per 
second.  In  the  two  centuries  during  which  their  proper 
motions  have  been  observed,  they  have  in  all  cases  moved  in 
sensibly  straight  lines  with  uniform  velocities.  Likewise, 
spectroscopic  determinations  of  motion  in  the  hne  of  sight 
give  no  evidence  of  anything  but  uniform  rectilinear  motion. 
These  statements  require  modification,  however,  in  the  case 
of  the  binary  stars  (Art.  283) . 


492    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  279 

There  is  no  doubt  that  the  paths  of  the  stars  eventually 
curve,  but  the  time  covered  by  our  observations  is  as  yet  far 
too  short  for  us  to  detect  these  deviations.  It  compares 
with  the  vast  intervals  required  for  the  stars  to  move  across 
the  sidereal  universe  as  one  tenth  of  a  second  compares  with 
the 'period  of  the  earth's  revolution  around  the  sun. 

The  first  question  that  springs  to  the  mind  is  whether  the 
stars  travel  in  sensibly  fixed  and  closed  orbits  similar  to  those 
of  the  planets,  or  move  on  indefinitely  throughout  the  region 
occupied  by  the  stars  without  ever  retracing  any  parts  of 
their  paths.  Since  observations  cannot  at  present  answer 
this  question,  the  reply  must  be  based  on  dynamical  considera- 
tions. There  is  clearly  no  central  mass  among  the  stars  and 
there  is  no  center  about  which  they  seem  to  be  distributed 
with  ansrthing  approaching  synmietry.  Moreover,  their 
motions  give  no  hint  that  they  are  moving,  even  temporarily, 
around  some  central  mass  or  point. 

The  con-elusion  is  inevitable  that  the  stars  describe  more 
or  less  irregular  paths,  in  the  course  of  time  probably  extend- 
ing into  all  parts  of  the  sidereal  system.  In  fact,  the  Galaxy 
was  hkened  by  Kelvin  to  a  great  gas  in  which  the  stars  cor- 
respond to  the  molecules.  When  they  are  far  apart  their 
mutual  attractions  are  inappreciable,  just  as  molecules  do 
not  interfere  with  the  motions  of  one  another  except  at  the 
times  of  colhsions.  •  If  two  stars  should  colUde  they  would 
probably  coalesce,  the  heat  generated  by  their  impact  chang- 
ing them  into  the  nebulous  state.  This  would  be  quite  dif- 
ferent from  an  elastic  rebound  of  molecules.  But  actual  col- 
hsions would  be  excessively  rare  and  near  approaches  would 
be  relatively  much  more  frequent.  A  near  approach  is 
dynamically  equivalent  to  an  obhque  impact  of  perfectly 
elastic  bodies,  as  is  illustrated  in  Fig.  170.  In  this  figure 
C  is  the  center  of  gravity  around  which  as  a  focus  the  two 
masses  (assumed  equal)  describe  hyperbolas.  It  is  easy  to 
see  that  the  motion  before  and  after  near  approach  is  similar 
to  that  of  two  elastic  spheres  colUding  a  little  to  the  right  of 


CH.  XIII,  279]        THE   SIDEREAL  UNIVERSE 


493 


their  respective  centers.  Consequently  there  are  some  good 
grounds  for  comparing  the  sidereal  system  to  a  vast  mass  ot 
gas. 

There  are,  however,  fundamental  differences  between  a 
gas  and  the  stellar  system.  In  a  gas  the  collisions  are  the 
important  events  in  the  history  of  a  molecule,  and  are  the 
only  appreciable  factors  which  influence  its  motion.  In  the 
stellar  system  the  near  approaches  of  a  given  star  to  some 
other  one  are  excessively  rare, 
and  the  attraction  of  the  whole 
system  is  the  primary  factor 
determining  the  motion  of  the 
individual  star.  Or,  more 
particularly,  a  molecule  in  a 
vessel  of  ordinary  gas  has 
thousands  of  millions  of  col- 
hsions  with  other  molecules 
per  second,  while  the  attrac- 
tion of  the  whole  mass  has  no 
appreciable  effect  on  its  mo- 
tion. But  in  the  sidereal 
system,  a  star  will  in  general 
travel  several  times  from  one 
of  its  visible  borders  to  the  Fig.  170. 
opposite  one  without  once 
passing  near  enough  to  an- 
other star  to  have  its  motion  radically  altered  by  the  latter, 
while  its  motion  is  controlled  by  the  attraction  of  the  whole 
mass  of  stars. 

It  is  difficult  to  realize  the  great  distances  which  separate 
the  stars  and  how  feeble  are  the  forces  with  which  they 
attract  one  another.  If  the  earth  were  at  rest,  it  would  fall 
toward  the  sun  less  than  one  eighth  of  an  inch  the  first  second. 
The  distance  of  the  relatively  near  star  Sirius  is  500,000 
times  as  great ;  and  in  spite  of  the  fact  that  its  mass  is  3.4 
■times  that  of  the  sun,  in  a  whole  year  it  would  give  the  sun 


-Near  approach  of  two 
stars  is  similar  to  an  oblique  col- 
lision of  elastic  bodies. 


494    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  279 

a  velocity  of  only  0.00007  of  an  inch  per  second.  Only 
after  900,000,000  years  at  the  present  distance  would  the 
relative  velocity  of  the  two  amount  to  one  mile  per  second. 
Long  before  such  an  immense  time  shall  have  elapsed  the 
sun  and  Sirius  will  be  far  separated  in  space. 

Now  consider  a  group  of  stars,  such  as  the  cluster  in  Taurus, 
traveling  through  the  stellar  system.  So  far  as  their  mutual 
interactions  on  one  another  are  concerned  the  result  is  the 
same  as  though  they  were  not  moving^with  respect  to  the 
other  stars.  In  their  motion  through  space  they  are  subject 
as  a  whole  to  the  changing  attractions  of  the  other  stars, 
and  individually  to  possible  close  approaches.  These  fac- 
tors may  be  considered  separately. 

The  Taurus  cluster  consists  of  39  (possibly  more)  stars 
which  occupy  a  space  whose  diameter  is  roughly  10  parsecs. 
From  the  high  luminosity  of  the  individual  members  of  the 
group  it  is  reasonable  to  suppose  that  they  have  large  masses, 
and  it  will  be  supposed  that  they  average  10  times  the  sun 
in  mass.  It  will  be  assumed  that  their  motions  are  such 
that  they  are  neither  simply  falhng  together  nor  scattering 
more  widely  in  space,  and  that  they  are  distributed  uniformly 
throughout  the  volume  which  they  occupy.  That  is,  it  is 
assumed  that  there  is  a  balance  (speaking  roughly)  between 
the  gravitational  forces  among  them  and  the  centrifugal 
forces  due  to  their  relative  motions.  With  these  data  and 
assumptions  their  maximum  velocities  with  respect  to  the 
center  of  gravity  of  the  group,  and  the  time  required  for  one 
of  them  to  move  from  one  border  of  the  group  to  the  oppo- 
site, can  be  computed. 

It  is  found  that  the  velocities  of  the  stars  of  the  group  with 
respect  to  their  center  of  gravity  will  always  be  less  than 
0.4  of  a  mile  per  second,  and  this  maximum  will  be  approached 
only  very  infrequently.  If  their  masses  are  comparable  to 
that  of  the  sun  instead  of  being  10  times  as  great,  the  veloci- 
ties relative  to  their  center  of  mass  will  always  be  less  than 
0. 13  of  a  mile  per  second.     Consequently,  the  internal  motions 


CH.  XIII,  279]        THE    SIDEREAL  UNIVERSE  495 

of  the  group  due  to  the  mutual  attractions  of  its  members 
will  always  be  small,  and  the  fact  that  at  present  the  stars 
are  moving  in  sensibly  parallel  lines  with  the  same  speed  does 
not  in  the  least  justify  the  conclusion  that  the  members  of 
the  cluster  are  in  any  sense  young.  It  is  also  found  that  the 
time  required  for  a  star  to  move  from  one  side  of  the  group 
to  the  other  under  the  attraction  of  all  the  stars  in  it  is 
25,000,000  years.  At  present  it  does  not  seem  safe  to  put 
any  time  limits  on  the  Ufe  of  a  star,  and  consequently  it 
may  be  supposed,  at  least  tentatively,  that  the  cluster  has 
been  in  existence  long  enough  for  the  stars  of  which  it  is 
composed  to  have  made  many  excursions  across  it.  The 
mutual  interactions  of  the  stars  have  a  tendency  to  make 
the  cluster  uniformly  spherical  with  the  stars  of  greatest 
mass  somewhat  condensed  toward  the  center.  The  approxi- 
mate sphericity  of  the  group  is  in  harmony  with  the  hypothe- 
sis that  it  is  very  old. 

It  remains  to  consider  the  effect  on  the  cluster  of  its  pas- 
sage through  star-strewn  space.  The  result  depends,  of 
course,  upon  the  star  density  of  the  region  which  it  traverses. 
It  has  been  seen  that  there  are  20  known  stars  within  5 
parsecs  of  the  earth.  It  is  not  imreasonable  to  suppose- 
that  there  are  10  other  stars  within  the  same  distance  of  the 
earth  which  are  at  present  unknown.  Under  the  assumption 
that  the  stars  are  scattered  uniformly  with  a  density  such 
that  there  are  30  within  a  sphere  whose  radius  is  5  parsecs, 
it  is  found  that,  on  the  average,  the  cluster  will  have  to  pass 
over  a  distance  of  5700  parsecs  in  order  that  at  least  one  of 
its  39  members  shall  pass  another  star  within  1000  times 
the  distance  from  the  earth  to  the  sun.  Since  the  cluster 
moves  at  the  rate  of  about  16  miles  per  second  with  respect 
to  the  stars  now  surrounding  it,  about  40,000  years  will  be 
required  for  it  to  describe  one  parsec ;  and  to  pass  'over 
5700  parsecs  will  require  more  than  200  million  years.  But 
5700  parsecs  is  probably  far  beyond  the  limits  of  the  visi- 
ble universe,  and  before  the  cluster  shall  have  traversed  any 


4%    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  279 

considerable  fraction  of  this  distance  the  attraction  of  the 
great  mass  of  stars  in  the  Galaxy  will  have  radically  altered, 
and  possibly  reversed,  its  motion. 

While  the  stars  of  the  cluster  pass  close  to  other  stars  only 
after  very  long  intei'vals,  they  are  continually  subject  to 
slight  disturbing  forces  which  affect  them  somewhat  un- 
equally. This  results  in  a  slight  tendency  to  scatter  the 
members  of  the  group.  One  might  be  tempted  to  conclude 
from  the  fact  that  it  is  still  very  coherent  that  its  age  should 
be  counted  in  hundreds  of  millions  of  years  at  the  most. 
But  it  is  impossible  to  determine  how  many  stars  once  be- 
longing to  it  have  been  torn  from  it  by  near  approaches  to 
other  stars,  or  how  many  of  the  smaller  original  stars  have 
been  thrown  to  its  borders  by  its  internal  interactions  and 
then  removed  by  the  differential  attractions  of  exterior 
bodies,  or  how  much  more  condensed  it  may  formerly  have 
been.  In  short,  no  certain  conclusions  respecting  the  age  of 
one  of  these  moving  clusters  can  be  drawn  from  the  proper- 
ties of  the  motion  of  their  members  at  present. 

It  is  now  possible  to  pass  to  the  consideration  of  the  whole 
sidereal  system.  The  star-streams  discovered  by  Kapteyn 
and  the  form  of  the  Galaxy  suggest  that  it  is  made  up  largely 
of  many  vast  star  clouds  which  move  at  least  approximately 
in  the  plane  of  the  Milky  Way.  There  is  a  general  tendency 
for  the  mutual  interactions  of  the  members  of  each  star 
cloud  to  reduce  it  to  the  spherical  or  symmetrically  oblate 
form.  Moreover,  the  stars  of  smaller  mass  gradually  acquire 
greater  velocities  at  the  expense  of  the  larger  stars,  just  as 
in  a  mixture  of  gases  of  molecules  of  different  weights  the 
lighter  ones  on  the  average  move  faster  than  the  heavier 
ones.  The  fact  that  the  individual  star  clouds  are  not 
spherical  would  argue  that  they  have  not  had  time  to  acquire 
the  symmetrical  form  of  equilibrium,  if  it  were  not  for  the 
fact  that  their  passage  through  and  near  to  other  star- 
clouds  may  occasionally  introduce  great  irregularities. 

But  all  the  star  clouds  which  together  constitute  the  Milky 


CH.  XIII,  279]        THE    SIDEREAL  UNIVERSE  497 

Way  may  be  considered  as  being  simply  a  much  larger  sys- 
tem. If  it  remains  isolated  from  all  other  systems,  it  will 
similarly  tend  toward  a  symmetrical  form.  Its  irregularities 
point  toward  the  conclusion  that  its  age  is  not  indefinitely 
great ;  and  this  would  be  a  necessary  conclusion  if  there  were 
not  the  possibility,  or  perhaps  even  probability,  of  the  exist- 
ence of  other  galaxies  beyond  our  own  near  which,  or  through 
which,  ours  passes  after  intervals  of  time  of  a  higher  order 
of  magnitude  than  any  so  far  considered.  These  famihes  of 
galaxies  may  be  units  in  still  larger  systems,  and  so  on  with- 
out Hmit.  Therefore  it  is  impossible  to  conclude  from  the 
irregularities  in  the  star  clouds  or  galaxies  that  they  have 
not  been  of  infinite  duration.  It  should  be  added  at  once 
that  most  astronomers  beheve,  chiefly  on  the  basis  of  the 
finite  amount  of  energy  of  the  stars,  that  they  have  not 
existed  for  an  infinite  time. 

While  it  has  not  been  possible  to  answer  the  more  ambi- 
tious questions  which  have  been  raised,  there  remain  others 
which  are  not  without  interest.  For  example,  suppose  that 
throughout  the  whole  region  occupied  by  the  stars  they  are 
as  numerous  as  they  are  near  the  sun ;  that  is,  that  there  are 
20  or  30  in  a  sphere  whose  radius  is  5  parsecs.  Suppose, 
further,  that  there  is  equilibrium  between  the  attractive  and 
centrifugal  forces.  So  far  as  these  assumptions  approximate 
the  truth,  there  is  a  relation  between  the  dimensions  of  the 
whole  stellar  system  and  the  mean  velocity  of  stars  at  its 
center,  for  the  velocities  depend  upon  the  star  density  and 
the  extent  of  the  region  which  they  occupy.  Inasmuch  as 
the  star  density  in  the  neighborhood  of  the  sun  and  the 
velocities  of  the  stars  have  been  determined  by  observa- 
tions, the  extent  of  the  whole  system  can  be  computed. 

The  solar  system,  which  is  far  from  the  borders  of  the 
Gala::^y,  will  be  supposed  to  be  approximately  at  its  center. 
The  mean  velocity  of  the  stars  near  the  sun  is  about  22  miles 
per  second.  This  fact  and  the  assumptions  which  have  been 
made  imply  that  the  radius  of  the  Galaxy  is  about  1100 
2k 


498    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  279 

parsecs  and  that  the  total  number  of  stars  in  it  is  260,000,000. 
Although  the  assumptions  are  not  in  exact  harmony  with 
the  facts,  it  is  beheved  that  these  results  are  of  the  correct 
order  of  magnitude.  And  under  the  same  assumptions  the 
time  required  for  a  star  to  pass  from  one  side  of  the  system 
to  the  opposite  is  approximately  200,000,000  years.  Since 
this  is  probably  less  than  the  age  of  the  earth,  our  sun  may 
have  traveled  in  geological  times  more  than  once  far  toward 
the  boundaries  of  the  stellar  system. 

Whatever  may  have  been  the  history  of  any  particular 
star,  these  results,  though  they  may  be  appreciably  in  error 
numerically,  imply  that  the  stars  have  undergone  consider- 
able mixing.  So  far  as  can  be  determined  at  present  this 
process  will  continue  in  the  future,  the  star  clouds  which 
form  the  Milky  Way  will  become  more  and  more  uniform 
and  the  motions  of  the  stars  more  and  more  chaotic,  the  stars 
of  smaller  mass  will  acquire  higher  velocities  than  the  larger 
ones,  at  rare  intervals  every  star  will  pass  near  some  other 
star,  and  possibly  at  intervals  of  time  of  a  higher  order  our 
Galaxy  will  encounter  other  galaxies  and  again  be  deformed 
and  made  irregular  by  them. 

280.  Runaway  Stars.  —  Since  the  average  radial  velocity 
of  a  large  group  of  stars  is  one  half  the  average  of  their  entire 
motions,  the  spectroscope  furnishes  the  average  speed  with 
which  the  stars  move.  The  average  velocity  of  the  stars 
near  the  sun  is  about  1.8  times  the  velocity  of  the  sun,  or 
22  miles  per  second.  This  is  7.5  astronomical  units  per  year, 
or  one  parsec  in  about  27,000  years. 

The  stars,  however,  do  not  all  move  with  even  approxi- 
mately the  same  velocity.  The  variations  in  their  speeds 
are  evidenced  both  by  their  proper  motions  and  by  their 
radial  velocities.  The  star  having  the  largest  known  proper 
motion,!  namely,  8  ".7  per  year,  is  the  sixth  in  Table  XVI, 

1  Professor  Barnard  has  just  (June,  1916)  found  an  eleventh-magnitude 
star  in  Ophiuchus  whose  annual  proper  motion  is  over  10" ;  its  parallax 
has  not  yet  been  measured. 


CH.  XIII,  280]        THE   SIDEREAL  UNIVERSE  499 

and  by  astronomers  is  known  as  C.  Z.  5  h.  243,  or  No.  243' 
in  the  fifth  hour  of  right  ascension  in  the  Cordoba  Zone 
Catalogue.  It  was  discovered  by  Kapteyn  in  1897  from  the 
measurement  of  plates  taken  by  Gill  and  Innes  at  the  Cape 
Observatory,  in  South  Africa.  Its  actual  velocity  is  170 
miles  per  second,  or  nearly  8  times  the  average  velocity  of 
the  stars.  The  star  known  as  1830  Groombridge  has  a 
proper  motion  of  7"  per  year.  Its  parallax,  which  is  not 
yet  accurately  known,  can  scarcely  exceed  0".l  and  its 
velocity  probably  exceeds  200  miles  per  second.  The  star 
61  Cygni  is  another  one  in  Table  XVI  which  moves  at  a  high 
speed,  though  its  velocity  is  exceeded  by  the  velocities  of 
quite  a  number  of  other  known  stars. 

The  stars  ha-ving  high  velocities  are  called  "  runaway 
stars  "  because,  unless  they  pass  very  near  other  stars  in 
their  journey  through  space,  they  will  escape,  like  molecules 
from  a  planet,  from  the  gravitative  control  of  the  stars  which 
constitute  the  Galaxy,  and  will  recede  from  them  forever. 
This  conclusion  is  inevitable  unless  the  total  mass  of  the 
sidereal  system  is  much  greater  than  has  hitherto  been  sup- 
posed. Even  if  the  extravagant  assumption  is  made  that 
there  are  1,000,000,000  stars,  each  as  massive  as  the  sun,  in 
a  spherical  space  whose  radius  is  1000  parsecs,  it  is  found  that 
a  star  moving  through  its  center  with  a  speed  exceeding  72 
miles  per  second  will  entirely  escape  from  the  system  unless, 
in  its  journey  toward  the  surface,  it  passes  near  at  least  one 
other  star  in  a  particularly  favorable  way  so  that  its  velocity 
is  much  reduced.  Since  the  probabiUty  of  such  a  near  ap- 
proach is  very  small,  we  are  forced  to  the  conclusion  that  these 
stars  with  high  velocities  are  only  temporary  members  of  our 
Gala'xy.  The  only  alternative  is  that  the  mass  of  the  sys- 
tem is  at  least  10  times  as  great  as  has  been  estimated. 

If  the  total  mass  of  the  stellar  system  is  greatly  in  excess 
of  the  estimates  which  have  been  made,  the  resulting  attrac- 
tive forces  are  greater  than  the  centrifugal  forces  due  to  the 
average  motions  of  the  stars,  and,  therefore,  the  stars  must 


500    AN   INTRODUCTION  TO  ASTRONOMY   [ch.  xm,  280 

be  on  the  whole  falhng  together.  That  is,  either  the  run- 
.away  stars  will  actually  escape  from  the  Galaxy  entirely,  or 
the  stellar  system  will  necessarily  become  more  and  more 
concentrated  under  the  mutual  gravitation  of  its  parts. 

The  question  of  the  origin  of  runaway  stars  at  once  arises. 
Either  they  have  come  in  from  beyond  our  Galaxy,  perhaps 
from  a  distant  one,  or  their  high  velocities  have  been  de- 
veloped within  our  stellar  system.  The  first  alternative  is 
certainly  possible  though  it  may  appear  at  first  to  be  im- 
probable, especially  in  view  of  the  enormous  time  required 
for  a  star  to  go  from  one  sidereal  system  to  another.  But 
these  stars  will,  in  most  cases,  permanently  leave  our  Galaxy, 
and  there  is  no  apparent  reason  why  stars  might  not  equally 
well  leave  other  galaxies. 

The  second  alternative  is  also  possible,  for  if  a  large  star 
and  a  small  star  pass  near  each  other  the  velocity  of  the  small 
one  may  be  greatly  increased.  A  series  of  favorable  close 
approaches  might  easily  produce  the  high  velocities  which 
are  observed.  The  process  is  closely  analogous  to  the  de- 
velopment of  high  velocities  in  exceptional  cases  in  a  mixture 
of  gases,  the  light  molecules  acquiring  the  highest  velocities. 
The  difficulty  in  the  case  of  the  stars  is  that  the  intervals 
between  close  approaches  are  so  long  that  the  process  de- 
mands starthng  lengths  of  time.  Perhaps  astronomers  in 
the  remote  future  will  be  able  to  determine  from  their 
greater  knowledge  regarding  the  masses  and  the  velocities 
bf  the  stars  something  respecting  the  length  of  time  during 
which  the  stars  of  the  stellar  system  have  been  subject  to 
their  mutual  attractions. 

281.  Globular  Star  Clusters.  —  Perhaps  the  most  won- 
derful objects  in  the  heavens  are  the  dense  globular  star 
clusters.  They  cover  portions  of  the  sky  generally  less  than 
30'  in  diameter,  that  is,  less  than  the  apparent  diameter  of 
the  moon.  The  brightest  of  them  appear  to  the  unaided 
eye  as  faint  fuzzy  stars,  but  a  large  telescope  shows  that  they 
are  made  up  of  thousands  of  stars.     The  most  splendid  of 


CH.  XIII,  281]        THE    SIDEREAL  UNIVERSE 


501 


these  objects  in  the  northern  sky  is  the  great  Hercules  cluster 
(Fig.  171),  also  known  to  astronomers  as  Messier  13,  in  which 


Fia.  171.  —  The  great  globular  star  cluster  in  Hercules  (M.   13).     Photo- 
graphed  by  Ritchey  with  the  40-inch  telescope  of  the  Yerkes  Observatory. 

Ritchey's  photograph,  taken  with  the  great  60-inch  reflector 
of  the  Mt.  Wilson  Solar  Observatory,  shows  more  than  50,000 
stars.  The  great  cluster  Omega  Centauri,  in  the  southern 
heavens,  is  even  a  more  wonderful  aggregation  of  suns. 


502    AN   INTRODUCTION   TO  ASTRONOMY    tea.  Am, -ol 

The  individual  stars  in  most  of  the  globular  clusters  are 
very  faint,  ranging  from  about  the  twelfth  magnitude  down 
to  the  Umits  of  visibility  with  the  instrument  employed. 
If  we  knew  the  distance  of  a  cluster,  we  could  determine  the 
luminosity  of  its  members  compared  to  the  sun.  Then  we 
could  answer  the  question  whether  the  stars  in  the  clusters 
are  great  suns  hke  our  own,  but  which  appear  faint  and 
crowded  together  only  because  of  their  immense  distance 
from  us,  or  whether  they  are  examples  of  an  evolution  in 
which  the  mass  is  distributed  among  a  very  large  number  of 
relatively  small  bodies.  It  is  not  possible  to  measure  directly 
the  parallaxes  of  the  globular  clusters,  and  their  probable 
distances  can  be  inferred  only  from  their  proper  motions. 
Unfortunately,  we  do  not  yet  have  any  positive  data  bearing 
on  the  problem  except  that  their  positions  in  the  sky  are 
sensibly  fixed.  This  can  only  mean  that  they  are  very  dis- 
tant, for  there  are  more  than  100  clusters  known,  and  it  is 
improbable  that  all  of  them  should  be  moving  in  the  same 
direction  as  the  sun  and  with  the  same  speed.  It  seems  to 
be  clear  from  their  apparent  fixity  on  the  sky  that  their  dis- 
tance is  at  least  100  parsecs  and  it  is  much  more  probable 
that  it  is  1000  parsecs.  At  the  distance  of  100  parsecs  the 
sun  would  be  a  ninth-magnitude  star,  wliile  at  1000  parsecs 
it  would  be  of  the  fourteenth  magnitude.  If  the  clusters 
are  at  the  smaller  distance,  their  members  are  much  less 
luminous  than  the  sun ;  if  at  the  greater,  they  are  comparable 
with  the  sun. 

The  problem  may  also  be  considered  in  the  reverse  order. 
That  is,  if  there  are  any  reasons  for  assuming  that  the  indi- 
vidual stars  in  the  clusters  are  comparable  to  the  sun  in 
luminosity,  or  related  to  it  in  any  definite  way,  then  their 
distances  can  be  computed.  The  stars  in  the  clusters  are 
individually  so  faint  that  theij  spectra  cannot  be  studied; 
but  valuable  information  concerning  the  character  of  the 
hght  they  radiate  can  be  obtained  by  photographing  them 
first  with  plates  sensitive  to  the  blue  and  then  to  the  red 


CH.  XIII,  281]        THE   SIDEREAL  UNIVERSE  503 

end  of  the  spectrum.  Such  work  has  been  carried  out  at 
the  Solar  Observatory  and  Shapley  finds  evidence  that  the 
stars  in  the  Hercules  cluster  are  like  the  giant  red  and  yellow 
stars,  such  as  Antares  and  Arcturus,  which  are  enormously 
more  luminous  than  the  sun.  If  this  conclusion  is  correct, 
the  distance  of  the  Hercules  cluster  is  of  the  order  of  10,000 
parsecs.  Perhaps  a  reasonable  summary  of  present  infor- 
mation would  be  that  globular  clusters  are  almost  certainly  - 
distant  much  more  than  100  parsecs,  and  that  their  distances 
probably  range  from  1000  to  10,000  parsecs. 

The  actual  dimensions  of  the  clusters  are  appalUng.  The 
distance  across  one  whose  apparent  diameter  is  30'  is  j^  of 
its  distance  from  the  earth,  or  probably  of  the  order  of  at 
least  10  parsecs.  If  50,000  stars  were  distributed  uniformly 
throughout  a  sphere  of  these  dimensions,  the  average  distance 
between  adjacent  stars  would  be  more  than  0.4  parsec,  or 
more  than  80,000  times  the  distance  from  the  earth  to  the 
sun.  It  is  seen  from  this  that,  although  the  globular  clusters 
are  somewhat  condensed  toward  their  centers,  the  actual 
distances  between  the  stars  of  which  they  are  composed  are 
enormous.  There  is  abundance  of  room  in  them  for  almost 
indefinite  motion  without  colHsion,  and  there  is  no  apparent 
reason  why  the  individual  stars  should  not  have  planets 
revolving  around  them. 

Dynamically,  the  globular  clusters  are  much  simpler  than 
the  Galaxy.  They  seem  to  have  arrived  at  an  approxi- 
mately fixed  state  of  symmetrical  distribution,  though,  of 
course,  the  individual  stars  are  in  ceaseless  motion  through 
them.  The  regularity  of  their  arrangement  implies  that  the 
process  of  mixing  has  been  in  operation  an  enormous  time, 
unless  indeed  they  started  in  this  remarkable  state.  It  is 
not  difficult  to  get  at  least  an  approximate  idea  of  the  time 
required  for  a  star  to  move  from  the  borders  to  the  center  of 
a  globular  cluster.  The  distribution  of  mass  in  a  cluster  is 
somewhere  between  condensation  entirely  at  the  center  and 
uniform  density.     In  the  first  case  the  force  varies  inversely 


504    AN   INTRODUCTION'  TO  ASTRONOMY  [ch.  xiii,  '^si 

as  the  square  of  the  distance  from  the  center,  and  in  the 
second,  it  varies  directly  as  the  distance  from  the  center. 
In  a  cluster  whose  radius  is  5  parsecs  and  which  contains 
50,000  stars,  each  having  the  mass  of  the  sun,  the  time  re- 
quired for  a  star  to  move  from  the  surface  to  the  center  in 
the  first  case  is  nearly  800,000  years,  and  in  the  second  is 
1,100,000  years.  The  actual  time  is  of  the  order  of  1,000,000 
years.  Since  thousands  of  these  excursions  would  be  neces- 
sary to  reduce  a  group  of  stars  with  considerable  irregulari- 
ties in  distribution  to  the  symmetrical  forms  observed,  the 
age  of  these  systems  must  be  enormous.  Only  a.  thousand 
excursions  from  the  periphery  to  the  center  and  back  would 
require  1,000,000,000  years.  It  is  improbable  that  this 
number  is  too  large  (it  may  be  many  times  too  small),  and 
it  follows  that  either  the  stars  exist  an  enormous  time  as 
luminous  bodies,  or  much  of  the  dynamical  evolution  of  the 
clusters  was  completed  before  the  star  stage,  if,  indeed, 
there  has  been  such  a  preceding  stage.  And  it  follows  further 
from  the  symmetry  of  the  clusters  that  for  at  least  hundreds 
of  miUions  of  years  they  have  not  passed  near  other  clusters. 
No  rapid  motions  of  stars  in  the  globular  clusters  are  to  be 
expected.  With  50,000  stars,  each  equal  to  the  sun  in  mass, 
distributed  uniformly  throughout  a  sphere  whose  radius  is 
5  parsecs,  the  velocity  of  a  permanent  member  of  the  group 
at  its  center  would  be  only  about  4  miles  per  second.  Since 
the  actual  clusters  have  strong  central  condensations,  the 
velocity  for  the  ideal  case  would  be  considerably  exceeded 
by  stars  near  their  centers.  Suppose  they  move  at  10  miles 
per  second  at  right  angles  to  the  line  of  sight.  At  a  distance 
of  1000  parsecs  they  would  move  with  respect  to  the  center 
of  the  cluster  only  one  second  of  arc  in  300  years.  Of  course, 
if  the  assumptions  as  to  the  distance  or  masses  are  wrong, 
the  result  will  be  wrong,  and,  besides,  a  certain  small  number 
of  the  stars,  especially  those  of  smallest  mass,  will  have 
motions  in  excess  of  the  mean  velocities.  But  it  is  improb- 
able that  relative  motions  of  the  members  of.  star  clusters 


CH.  XIII,  282]        THE    SIDEREAL  UNIVERSE  505 

1 

will  be  large  enough  in  any  case  to  be  observable  inside  of 
several  decades. 

XXIII.    QUESTIONS 

1.  Prove  that,  in  Pig.  168,  Z.  EiSEi  -  Z  EiS'Ei 

=  Z  SEiS'  -  Z  SEiS'. 

2.  Suppose  there  are  30  stars  within  5  parseos  of  the  sun ;  what  is 
the  average  distance  between  adjacent  stars  ? 

3.  Draw  a  diagram  to  prove  that  Herschel's  observations,  Art. 
274,  are  explained  by  the  conclusion  which  he  drew.  If  this  conclu- 
sion is  denied,  what  other  must  be  accepted  ? 

4.  If  an  angle  of  1".0  can  be  measured  with  an  error  Hot  exceeding 
10  per  cent,  how  small  a  parallax  can  be  determined  with  this  degree 
of  accuracy  by  the  method  of  Art.  275  in  100  years? 

6.  Show  by  a  diagram  that  if  two  stars  are  moving  in  parallel 
Unes,  then  the  great  circles  in  which  they  apparently  move,  as  seen 
from  the  earth,  intersect  in  a  point  whose  direction  from  the  earth  is 
the  direction  in  which  the  stars  move  (Art.  277). 

6.  Since  the  velocity  of  our  sun  is  somewhat  below  the  average  of 
the  velocities  so  far  measured,  what  are  the  probabihties  of  the  rela^ 
tion  of  its  mass  to  the  masses  of  the  observed  stars  ? 
^  7.  If  the  radius  of  the  Galaxy  is  1100  parsecs  (end  of  Art.  279), 
how  long  would  it  tak^  the  sun  at  its  present  speed  to  pass  from  the 
center  of  the  sidereal  system  to  its  borders  ? 

8.  If  the  velocity  of  the  star  1830  Groombridge  is  200  miles  per 
second  and  remains  constant,  how  long  will  be  required  for  it  to 
recede  to  a  distance  from  which  our  Galaxy  will  appear  as  a  hazy 
patch  of  light  1°  in  diameter?  ' 

9.  If  there  are  many  galaxies,  and  if  the  distances  between  them 
compare  to  their  dimensions  hke  the  distances  between  the  stars 
compare  to  the  dimensions  of  the  stars,  how  long  will  be  required  for 
1830  Groombridge  to  go  from  our  Galaxy  to  another  ? 

III.     The  Stabs 

282.  Double  Stars.  -^  A  few  double  stars  have  been  known 
almost  since  the  invention  of  the  telescope,  but  William 
Herschel  was  the  first  astronomer  to  search  for  them  sys- 
tematically and  to  measure  the  distances  and  the  directions 
of  their  components  from  one  another.  His  purpose  in  meas- 
uring them  was  to  determine  the  parallax  of  the  nearest  ones 


506    AN  INTRODUCTION   TO  ASTRONOMY  [ch.  xiii,  282 

(Art.  272),  for  he  assumed,  perhaps  unconsciously,  that  the 
sun  is  a  typical  star,  and  that  when  two  stars  are  apparently 
in  about  the  same  direction  from  the  earth,  one  is  simply 
farther  away  than  the  other. 

Herschel  found  a  large  number  of  double  stars  whose  com- 
ponents were  apparently  separated  by  a  few  seconds  of  arc 
at  the  most.  A  discussion  of  the  probability  of  there  being 
such  a  large  number  of  stars  so  nearly  in  lines  passing  through 
the  earth  would  have  shown  him  that  their  apparent  prox- 
imity could  not  be  accidental.  He  reached  the  same  result 
in  a  few  years,  for  his  observations  showed  him  in  a  con- 
siderable number  of  cases  that  the  two  components  were 
revolving  around  their  center  of  gravity.  That  is,  instead 
of  all  stars  consisting  of  single  primary  bodies  accompanied 
by  famiUes  of  planets,  there  are  many  which  are  twin  suns 
of  approximately  equal  mass  and  dimension.  So  far  as  we 
know,  they  may  or  may  not  have  planetary  attendants,  for 
such  small  objects  shining  entirely  by  reflected  Ught  would 
be  beyond  the  range  of  our  telescopes  even  if  they  were  a 
thousand  tirhes  more  powerful  than  any  yet  constructed. 

The  names  that  stand  out  most  prominently  in  the  double- 
star  astronomy  of  the  nineteenth  century  are  WilUam 
Struve,  Dawes,  John  Herschel,  and  Burnham.  In  Bum- 
ham's  great  catalogue  of  double  stars  the  observations  and 
descriptions  of  about  13,000  of  these  objects  are  given.  New 
ones  are  constantly  being  discovered,  though  the  northern 
heavens  have  now  been  very  thoroughly  examined  with 
powerful  telescopes.  At  the  Licl^  Observatory  a  survey  of 
the  whole  heavens  to  at  least  —14°  declination  was  begun 
by  Hussey  and  Aitken  and  completed  by  Aitken.  All  old 
pairs  with  a  separation  not  exceeding  5"  of  arc  were  observed, 
and  4300  new  pairs  were  discovered  within  the  same  hmits. 
On  using  a  definition  of  double  star  which  excludes  all  wider 
pairs  except  in  the  case  of  bright  stars,  Aitken  found  that 
there  are  5400  of  these  objects  not  fainter  than  the  ninth 
magnitude  north  of  the  celestial  equator.     This  means  that 


CH.  xm,  283]        THE   SIDEREAL  UNIVERSE  507 

at  least  one  star  in  18  of  those  not  fainter  than  the  ninth 
magnitude  is  a  double  which  is  visible  with  the  36-inch 
telescope  of  the  Lick  Observatory.  Of  these  stars,  2206 
have  an  apparent  angular  separation  not  greater  than  1", 
and  only  200  are  separated  by  more  than  5".  A  very  in- 
teresting fact  is  that,  compared  to  the  whole  number  of  stars 
of  the  same  brightness,  double  stars  seem  to  be  somewhat 
more  numerous  in  the  Milky  Way  than  near  its  poles. 
Moreover,  the  average  separation  of  the  stars  of  the  spectral 
class  to  which  the  sun  belongs  is  considerably  greater  than  in 
those  of  .the  so-called  earlier  types  which  include  the  blue  stars. 

There  are  doubtless  some  cases  in  which  the  components 
of  a  double  star  are  at  different  distances  and  simply  in  nearly 
the  same  direction  from  the  observer.  But  in  general  they 
form  physical  systems  which  revolve  around  their  centers  of 
gravity  in  harmony  with  the  law  of  gravitation,  and  these 
pairs  are  called  binaries.  According  to  the  law  of  probability, 
essentially  all  of  the  5400  double  stars  in  Aitken's  Ust  must 
be  binaries,  for  only  very  rarely  would  two  stars  be  acci- 
dentally so  nearly  in  the  same  direction  from  us. 

283.  The  Orbits  of  Binary  Stars.  —  The  stars  in  all  cases 
are  so  remote  from  us  that  the  components  of  a  binary  sys- 
tem cannot  be  seen  as  separate  stars  unless  they  are  a  great 
distance  apart.  But  when  the  components  of  a  binary  pair 
are  far  from  each  other,  their  period  of  revolution  is  long, 
and  observations  must  therefore  extend  over  many  years 
in  order  to  furnish  data  for  the  computation  of  their  orbits. 
Those  binary  stars  which  were  first  discovered  and  which 
have  been  longest  imder  observation  are  not  very  close 
together,  and,  while  in  many  cases  it  is  now  certain  from 
direct  observational  evidence  that  they  form  physical  sys- 
tems, there  are  only  40  or  50  in  which  the  observed  arcs  are 
long  enough  to  define  the  orbits  with  any  degree  of  precision. 
In  1896  See  pubUshed  the  orbits  of  40  of  the  best-known 
binary  stars. 

The  periods  of  known  visual  binary  stars  range  from  5.7 


508    AN   INTRODUCTION   TO  ASTRONOMY  [ch.  xiii,  283 

years,  for  Delta  Aquilse,  to  hundreds  and  probably  thousands 
of  years.  The  planes  of  their  orbits  are  inclined  at  all 
angles  to  the  line  joining  them  with  the  earth,  so  that,  as  a 
rule,  we  see  their  orbits  in  projection.  Indeed,  the  orbit  of 
42  ComsB  Berenices  is  sensibly  edgewise  to  us.  One  of  the 
most  interesting  things  about  the  orbits  of  binaries  is  that 
they  are  generally  considerably  eccentric.  In  the  40  orbits 
in  See's  hst  the  average  eccentricity  was  0.48,  or  twelve  times 
that  of  the  planetary  orbits.  The  orbit  of  the  binary  star 
Gamma  Virginis  has  an  eccentricity  of  0.9,  and  therefore  the 
greatest  distance  of  the  two  members  of  this  pair  from  each 
other  is  19  times  their  least  distance. 

284.  Masses  of  Binary  Stars.  —  The  masses  of  those 
planets  which  have  satellites  are  found  from  the  periods  and 
distances  of  their  respective  sateUites  (Art.  154).  The 
masses  of  Mercury  and  Venus  are  found  from  their  attrac- 
tions for  other  bodies,  especially  comets.  The  masses  of 
celestial  bodies  are  found  only  from  their  attraction  for  other 
bodies.  It  is  evident,  therefore,  that  the  mass  of  a  single  star 
remote  from  all  other  visible  bodies  cannot  be  found.  But 
when  the  dimensions  of  the  orbit  and  the  period  of  revolu- 
tion of  a  binary  pair  are  known,  their  combined  mass  can  be 
computed  just  as  the  mass  of  a  planet  is  computed. 

The  periods  of  binary  stars  are  determined  by  direct 
observations  of  their  apparent  positions.  The  dimensions 
of  the  orbit  of  a  binary  pair  can  be  determined  from  their 
apparent  distance  apart  and  their  distance  from  the  earth. 
The  chief  difficulty  lies  in  the  problem  of  finding  their  parallax, 
for  only  a  small  number  of  stars  are  within  measurable  dis- 
tance from  the  sun. 

Those  binary  stars  whose  periods  and  distances  are  known 
with  sufficient  approximation  to  make  the  mass  determina- 
tions of  value  are  given  in  Table  XVIII.  The  masses  of  all 
those  whose  parallaxes  are  less  than  0".2  are  subject  to  some 
uncertainty,  and  the  probable  error  is  great  if  the  parallaxes 
are  less  than  0".l. 


CH.  XIII,  284]       THE   SIDEREAL  UNIVERSE 


509 


Table  XVIII 


Stab 

Par- 

Period 

Semi- 

Com- 

BINED 

Luminos- 

allax 

axis 

Mass 

ity 

a  Centauri    .     .     . 

0.76 

81.2 

23.3 

1.9 

2.0 

Sirius 

0.38 

48.8 

20.0 

3.4 

48.0 

Prooyon  .... 

0.32 

39.0 

10.4 

0.7 

9.7 

rj  CassiopeisB     .     . 

.0.20 

300.  (?) 

47.4 

1.2 

1.4 

70  Ophiuchi      .     . 

0.17 

88.4 

26.8 

2.5 

1.2 

02  Eridani    .     . 

0.17 

180.0 

28.2 

0.7 

0.8 

Bradley  2388 

0.13 

45.8 

8.2 

0.3 

1.0 

85  Pegasi 

0.11 

26.3 

7.7 

0.7 

'0.8 

^  Hereulis    .     .     . 

0.10 

34.5 

13.5 

2.1 

11.4 

K  Pegasi  .... 

0.08 

11.4 

3.7 

0.4 

3.1 

m  Bootis      .     .     . 

0.05 

200.  (?) 

21.5 

0.2 

0.7 

In  this  table  the  periods  are  given  in  years,  the  semi-axes  in 
terms  of  the  earth's  distance  from  the  sun,  the  combined 
mass  in  terms  of  the  sun's  mass,  and  the  luminosity  in  terms 
of  the  sun's  luminosity  at  the  same  distance. 

Perhaps  the  most  interesting  thing  brought  out  by  the 
table  is  that  the  masses  of  all  of  these  stars  are  comparable 
to  that  of  the  sun,  and,  with  the  exception  of  Sirius,  their 
luminosities  do  not  differ  greatly  from  that  of  the  sun. 
But  there  are  not  enough  pairs  of  stars  in  the  table  to  justify 
any  very  positive  general  conclusion. 

If  the  orbits  of  each  of  the  two  components  of  a  binary 
star  with  respect  to  their  center  of  gravity  are  known,  their 
separate  masses  can  be  computed.  The  problem  of  deter- 
mining the  orbits  of  two  stars  with  respect  to  the  center 
of  mass  of  their  system  is  very  difficult  because  their  motions 
with  respect  to  neighboring  sj;ars,  or  fixed  reference  fines, 
must  be  measured.  In  only  a  few  cases  are  the  results  at 
present  reliable.  The  discussions  of  Lewis  Boss  led  him  to 
the  conclusion  that  probably  in  all  cases  the  brighter  star 
is  the  more  massive,  a  result  which  is  contrary  to  that  which 
was  sometimes  found  in  earlier  investigations. 


510    AN   INTRODUCTION   TO  ASTRONOMY  [ch.  xiii,  285 

285.  Spectroscopic  Binary  Stars.  —  The  spectroscope  has 
contributed  very  important  results  to  the  study  of  binary 
stars.  Its  application  depends  upon  the  fact  that  it  enables 
the  observer  to  determine  whether  a  source  of  light  is  ap- 
proaching or  receding  (Art.  226).  Suppose  the  plane  of 
motion  of  a  binary  system  passes  through  the  earth,  as  is 
represented  in  Fig.  172.  When  the  stars  are  in  the  positions 
A  and  B,  one  is  receding  from,  and  the  other  is  approaching 
toward,  the  earth.  If  they  have  similar  spectra,  the' spec- 
trum of  the  combined  pair  will  consist  of  double  lines  (Fig. 

A 


TO  EARTH 


B 

Fig.  172.  —  Orbit  of  a  speutroscopic  binary  star. 

173),  for  the  lines  from  one  will  be  shifted  toward  the  red 
while  the  lines  from  the  other  will  be  displaced  toward  the 
violet.  When  the  stars  have  made  a  quarter  of  a  revolution 
around  their  center  of  gravity  0  and  have  arrived  at  A' 
and  B',  the  lines  will  not  be  displaced  because  the  stars  are 
neither  approaching  toward  nor  receding  from  the  observer. 
After  another  quarter  of  a  revolution  they  will  be  double 
again  because  A  will  be  approaching  and  B  receding. 

The  data  furnished  in  this  way  by  the  spectroscope  are 
very  important  because,  in  the  first  place,  the  separation  of 
the  lines  determines  the  relative  velocity  of  the  stars  in  their 
orbits.    This  is  true  whether  the  system  as  a  whole  is  sta- 


GH.  XIII,  285]       THE   SIDEREAL  UNIVERSE 


511 


tionary  ,with  respect  to  the  earth,  as  has  so  far  been  tacitly- 
assumed,  or  is  moving  in  the  line  of  sight.  The  period  is 
also  given.  The  period  and  velocity  furnish  the  dimensions 
of  the  orbit  and  consequently  the  total  mass  of  the  binary 
system. 

If  the  two  stars  of  the  binary  are  very  unequal  in  lumi- 
nosity, the  spectrum  of  the  fainter  one  will  not  be  obtained, 
but  the  spectral  lines  of  the  brighter  one  will  be  shifted 
alternately  toward  the  red  and  violet  ends  of  the  spectrum. 


Fig.  173.  —  Spectrum  of  Mizar,  showing  double  lines  above  and  single  lines 
below  (period  20.5  days).     {Frost;   Yerkes  Observatory.) 


The  period  is  given  in  this  case,  but  only  the  velocity  of  the 
brighter  star  with  respect  to  the  center  of  gravity  of  the 
system  is  known.  Since  the  orbit  of  one  star  with  respect 
to  the  other  is  necessarily  larger  than  the  orbit  of  the  brighter 
one  with  respect  to  the  center  of  gravity  of  the  two,  the  mass 
computed  in  this  case  will  always  be  too  small. 

It  has  so  far  been  assumed  that  the  plane  of  motion  of 
the  binary  star  passes  through  the  earth.  This  condition 
is  reaUzed  only  very  exceptionally,  and  indeed  is  not  neces- 
sary for  the  application  of  the  method.  If  the  plane  of 
motion  does  not  pass  exactly  through  the  earth,  the  meas- 
ured radial  velocity  is  only  a  fraction  of  the  whole  velocity, 


512    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  285 

and  the  size  of  the  orbit  and  mass  of  the  system  based  on  it 
are  both  too  small.  Since  the  planes  of  the  orbits  of  binary 
stars  may  have  any  relation  to  the  observerj  the  measured 
radial  velocities  are  in  general  smaller  than  the  actual 
velocities;  on  the  average  the  former  are  0.63  of  the 
latter.  On  the  average  the  calculated  masses  are  about  60 
per  cent  of  the  true  masses. 

The  spectroscope  is  particularly  valuable  in  the  study  of 
binary  stars  because  it  is  not  necessary  that  they  should  be 
near  enough  to  appear  as  visual  binaries.  The  only  requi- 
site is  that  they  shall  be  bright  enough  (above  the  eighth 
magnitude  with  present  instruments)  to  enable  astronomers 
to  photograph  their  spectra  in  a  reasonable  time.  With 
very  few  exceptions  the  spectroscopic  binaries  so  far  known 
are  not  also  visual  binaries.  A  second  advantage  of  the 
spectroscope  is  that  it  furnishes  at  the  same  time  lower 
limits  for  the  orbital  dimensions  and  masses  of  the  stars. 

The  first  known  spectroscopic  binary  was  discovered  by 
E.  C.  Pickering  at  the  Harvard  Observatory,  in  1889,  when 
it  was  found  that  the  spectrum  of  Mizar  (f  Ursae  Majoris) 
consisted  of  alternately  double  and  single  lines  (Fig.  173). 
Mizar  is  a  visual  double  star,  but  the  double  lines  belong  to 
a  single  component  of  the  visual  pair.  The  visual  pair  prob- 
ably are  revolving  around  their  center  of  gravity,  but  their 
distance  apart  is  so  great  that  their  period  of  revolution  is 
very  long  and  their  motions  are  too  slow  to  be  measured 
with  the  spectroscope. 

The  first  spectroscopic  binary  in  which  one  of  the  com- 
ponents is  dark  was  discovered  by  Vogel,  at  Potsdam,  in 
1889.  He  found  that  the  lines  in  the  spectrum  of  Algol, 
the  well-known  variable  star,  shift  alternately  toward  the 
red  and  blue  ends  of  the  spectrum  with  the  same  period  as 
that  of  its  variability  (2  d.  20  h.  49  m.).  This  confirmed 
the  theory  that  this  star  varies  in  brightness  because  a  rela- 
tively dark  one  revolves  around  it  and  partially  echpses  it 
at  each  revolution.     The  star  Mu  Orionis  has  the  short  period 


CH.  XIII,  286]       THE    SIDEREAL  UNIVERSE  513 

of  4.45  days,  and  the  displacements  of  its  spectral  lines  are 
considerable  (Fig.  174). 

In  1898  only  13  spectroscopic  binary  stars  were  known. 
By  1905  the  number  had  increased  to  140  pairs,  6  of  which 
were  also  visual  binaries.  When  Campbell  published  his 
second  catalogue  of  spectroscopic  binaries  in  1910,  there  were 
306  known  pairs.  In  19  cases  the  spectra  of  both  stars  had 
been  measured,  and  from  the  absolute  displacements  of  each 
set  of  hues  their  relative  masses  had  been  determined.  With 
one  possible  exception  the  brighter  stars  of  the  systems  are 


( 


I1III^IWM—«1W,WI|PPP 


III 


Fig.   174.  —  Spectra  of  Mu  Orionis  (Frost;  Yerkes  Observatory). 

the  more  massive.  The  larger  stars  kre  generally  less  than 
twice  as  massive  as  the  smaller.  Of  course,  the  difference  is 
probably  much  greater  in  those  cases  where  the  spectrum  of 
the  smaller  star  is  too  faint  to  be  observed. 

286.  Interesting  Spectroscopic  Binaries.  —  Mizar.  As 
has  been  stated,  the  brighter  component  of  Mizar  was  the 
first  spectroscopic  binary  discovered.  The  later  work  of 
Vogel  showed  that  its  period  is  about  20.5  days,  from  which 
it  follows  in  connection  with  the  dimensions  of  its  orbit 
(22,000,000  miles. betwecHL. the  two  components)  that  the 
mass  of  the  system  is  at  least  four  times  that  of  the  sun. 
The  spectra  of  botkstars  are  present,  and  their  equal  displace- 
2l 


514    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  iHs6 

ment  proves  that  the  masses  of  the  two  components  are 
sensibly  equal.  The  center  of  gravity  of  the  system  is  ap- 
proaching the  solar  system  at  the  rate  of  about  9  miles  per 
second.  In  1908  Frost  and  Lee  found  that  the  other  com- 
ponent of  Mizar  is  also  a  spectroscopic  binary  of  the  type 
in  which  the  spectrum  of  only  one  star  of  the  pair  is  visible. 
In  1908  Frost  announced  that  Alcor  is  a  spectroscopic  binary 
of  short  period  in  which  both  spectra  are  observable.  There- 
fore Mizar  is  a  visual  double  each  of  whose  components  is  a 
spectroscopic  binary,  and  the  neighboring  Alcor  is  also  a 
binary. 

Spica.  One  of  the  earliest  known  spectroscopic  binaries 
is  the  first-magnitude  star  Spica  whose  spectral  hues  were 
found  to  vary  by  Vogel  in  1890.  The  spectrum  of  the 
fainter  component  has  also  been  observed.  The  period  of 
the  pair  is  4  days,  their  mean  distance  from  each  other  is 
about  11,000,000  miles,  and  their  masses  (neglecting  the  pos- 
sible reduction  due  to  the  inclination  of  their  orbit)  are 
respectively  9.6  and  5.8  times  that  of  the  sun.  This  system 
is  receding  from  the  sun  at  about  1.2  miles  per  second. 

Capella.  The  first-magnitude  star  Capella  is  a  spectro- 
scopic binary,  the  spectra  of  both  stars  being  visible,  in  which 
the  period  is  104  days  and  the  mean  distance  (possibly  much 
reduced  by  the  incKnation  of  the  plane  of  the  orbit)  about 
50,000,000  miles.  With  these  data  the  masses  of  this  pair 
are  found  to  be  at  least  1.2  and  0.9  that  of  the  sun.  This 
orbit  has  a  very  small  eccentricity.  These  stars  are  re- 
ceding from  the  solar  system  at  the  rate  of  nearly  20  miles 
per  second.  The  parallax  of  Capella  has  been  investigated 
with  the  utmost  care  by  Elkin,  who  found  for  it  0".09,  cor- 
responding to  a  distance  of  11  parsecs.  At  that  distance 
the  sun  would  be  only  ^  as  bright  as  Capella,  or  approxi- 
mately of  the  fifth  magnitude.  Since  the  spectrum  of 
Capella  is  almost  exactly  the  same  as  that  of  the  sun,  which 
naturally  leads  to  the  conclusion  that  the  temperature  and 
surface  brightness  of  Capella  are  approximately  equal  to 


CH.  xiii,  287]       THE    SIDEREAL  UNIVERSE  615 

those  of  the  sun,  it  seems  probable  that  the  or,bit  of  the  pair 
is  so  inclined  that  the  computed  masses  are  much  too  small. 

Polaris.  The  pole  star  has  two  darker  companions  dis- 
covered spectroscopically  by  Campbell  in  1889.  One  is  very 
close  to  the  bright  star  and  revolves  around  it  in  a  period  of 
a  Uttle  less  than  4  days,  while  the  second  companion  is  much 
more  distant  and  requires  about  12  years  to  complete  a 
revolution.  These  stars  are  all  quite  distinct  from  the  faint 
telescopic  companion  to  Polaris. 

Alpha  Centauri.  Alpha  Centauri  is  at  the  same  time  a 
visual  and  a  spectroscopic  binary.  Moreover,  its  parallax 
has  been  very  accurately  determined  by  direct  means,  so 
that  the  actual  distance  of  the  components  from  each  other 
and  their  masses  can  be  determined  (Table  XVIII).  Since 
the  same  results  can  be  determined  spectroscopically,  their 
comparison  affords  a  valuable  check  on  the  accuracy  of  the 
results.  The  spectroscopic  data  were  obtained  by  Wright 
at  the  branch  of  the  Lick  Observatory  in  South  America, 
and  the  results  obta^ined  from  them  agree  almost  exactly 
with  those  based  on  other  methods.  But  the  spectroscope 
gives  the  additional  fact,  which  cannot  be  determined  other- 
wise, that  Alpha  Centauri  is  approaching  the  sun  at  the  rate 
of  13.8  miles  per  second. 

287.  Variable  Stars.  —  A  star  whose  brightness  changes 
is  said  to  be  a  variable.  The  first  known  variable,  Omicron 
Ceti,  was  discovered  by  Fabricius  in  1596.  The  variability 
of  Algol  was  definitely  announced  by  Goodricke  in  1783, 
though  it  seems  to  have  been  noticed  a  century  earlier. 
The  following  year  he  recorded  the  variability  of  Beta  Lyrse. 
But  variable  stars  were  not  discovered  in  any  considerable 
numbers  until  toward  the  close  of  the  nineteenth  century. 
Now  more  than  3000  of  these  objects  are  known  in  addition 
to  those  which  have  been  found  in  considerable  numbers  in 
some  of  the  globular  star  clusters.  Some  of  them  vary  regu- 
larly and  periodically,  with  periods  ranging  from  less  than  a 
day  to  more  than  two  years ;  others  vary  irregularly  with- 


516    AN   INTRODUCTION   TO  ASTKUin  ^jivx  i    i^n.  ^. 


out  any  apparent  rule  or  order.  Some  flash  out  brilliantly 
for  a  short  time  and  then  sink  back  more  slowly  into  per- 
manent oblivion.  It  is  certain  that  the  brightness  of  every 
star  varies  slowly  because  of  its  changing  distance  from  the 
sun,  if  for  no  other  reason,  but  there  is  no  observational 
evidence  of  a  change  for  this  reason. 

Variable  stars  are  classified  according  to  the  peculiarities 
of  their  hght  changes,  and  the  principal  types  are  enumerated 
in  the  following  articles.  It  must  be  remembered,  however, 
that  variable  stars  are  strange  objects  which  present  nu- 
merous exceptions  to  all  rules. 

288.  Eclipsing  Variables.  —  If  the  plane  of  the  orbit  of  a 
binary  pair  passes  very  nearly  through  the  earth,  the  stars 

partially  or  to- 
tally eclipse  each 
other  every  time 
they  are  in  a  line 
with  the  earth. 
If  one  of  the  two 
is  a  dark  star  and 
nearly  as  large  as 
the  bright  one,  it 
is  clear  that  the 
light    received 

Fig.   175.  —  Light  curve  of  typical  eclipsing  variable    from  the  pair  will 

remain  constant 
except  when  the  brighter  star  is  eclipsed.  As  the  dark  star 
begins  to  eclipse  the  brighter  one,  the  light  diminishes  very 
rapidly  until  the  time  of  greatest  obscuration,  after  which  as 
a  rule  the  star  rapidly  regains  its  normal  brightness.  How- 
ever, in  some  cases  the  dark  star  is  very  large  so  that  the 
eclipse  persists  for  a  considerable  time,  and  then  the  variable 
remains  at  minimum  for  a  few  minutes  or  possibly  a  few 
hours. 

The  variability  in  the  brightness  of  a  star  is  represented 
by  a  curve.     In  Fig.  175  the  curve  for  a  typical  eclipsing 


100 
/.iS 
I.SO 
175 
200 
ZJ5 

SJS 

a 

l> 

a 

« 

a 

'\ 

( 

' 

' 

. 

oArs         5             Lo             u             Z.O            gs            so             S.S 

CH.  XIII,  aj8J       THE    SIDEREAL  UNIVERSE  517 

variable  is  given.  The  time  is  marked  off  along  the  hori- 
zontal axis  and  the  brightness  of  the  star  is  proportional  to 
the  distance  of  the  curve  above  t'his  axis.  The  parts  marked 
a  give  the  brightne^  when  the  star  shines  undimmed  by  an 
eclips?,  the  points  h  are  where  the  light  begins  to  wane  as 
the  eclipse  commences,  and  the""points  c  indicate  the  bright- 
ness at  the  moment  of  greatest  obscuration.  If  the  fainter 
star  is  somewhat  luminous  instead  of  being  entirely  dark, 
there  will  be  a  secondary  and  less  pronounced  minimum. 

The  typical  eclipsing  variable  in  which  one  component  is 
dark  is  Algol  (Beta  Persei),  whose  light  curve  is  essentially 
the  same  as  that  given  in  Fig.  175.  About  100  stars  of  this 
type  are  known,  and  they  are  often  called  Algol  variables. 
They  are  characterized  by  the  shortness  of  their  periods, 
many  of  which  are  less  than  5  days  and  only  12  of  which 
are  longer  than  10  days,  and  by  the  regularity  of  their  light 
curves.  Doubtless  the  explanation  of  their  short  periods 
is  that  when  the  two  stars  are  far  apart  they  do  not  eclipse 
one  another,  even  partially,  unless  the  plane  of  their  motion 
passes  very  exactly  through  the  earth. 

Eclipsing  variables  are.  necessarily  spectroscopic  binary 
stars.  It  increases  our  confidence  in  both  the  methods  and 
the  interpretations  to  find  that  the  data  obtained  in  the 
two  distinct  ways  are  perfectly  in  accord.  It  is  not  to  be 
inferred  from  this  that  the  data  are  coextensive.  The  spec- 
troscope furnishes  the  velocity  and  therefore  the  dimensions 
and  mass  of  the  system,  especially  when  both  stars  are  lumi- 
nous. From  the  duration  of  the  eclipses  the  dimensions  of 
the  stars  can  be  found.  Since  their  masses  are  known,  their 
densities  can  then  be  computed.  It  has  been  found  by 
Russell,  Shapley,  and  other  astronomers  that  the  mean  den- 
sity of  the  variable  stars  for  which  there  are  sufficient  obser- 
vational data  is  about  one  eighth  that  of  the  sun.  This  is  a 
remarkable  result  in  view  of  the  fact  that  usually  one  of  the 
pair  is  very  liark,  and,  according  to  current  doctrine,  in  a 
condensed  state  approaching  extinction.     It  should  be  added 


518    AN   INTRODUCTION   TO  ASTRONOMY    [ch.  xin,  ^o 

that  in  the  case  where  there  is  a  single  minimum  the  result 
depends  upon  an  assumption  as  to  the  relative  densities  of 
the  components,  and  consequently  may  be  considerably  in 
error. 

The  period  of  Algol  is  2  d.  20  h.  48  m.  55  s.  It  is  normally 
a  stay  of  the  second  magnitude,  but  at  the  time  of  eclipse  it 
loses  five  sixths  of  its  light.  In  1889  Vogel  discovered  that 
it  is  a  spectroscopic  binary.  He  found  that  the  combined 
mass  of  the  system  is  two  thirds  that  of  the  sun,  the  bright 
star  has  twice  the  mass  of  the  darker  one,  the  distance  be- 
tween their  centers  is  about  3,000,000  miles,  the  diameters 
of  the  stars  are  about  1,000,000  and  800,000  miles,  and  their 
density  is  about  one  fourth  that  of  the  sun.  Schlesinger 
found  that  for  the  similar  system  Delta  Librae  the  density  is 
also  one  fourth  that  of  the  sun. 

There  are  several  variations  from  the  normal  Algol 
variable.  In  one  the  stars  are  of  unequal  size  and  both 
bright.  Then  each  eclipses  the  other,  but  the  loss  of  fight 
is  different  in  the  two  ecUpses,  and  the  light  curve  has  two 
minima  of  different  depths.  There  are  often  irregularities 
which  have  not  yet  been  explained.  Sometimes  the  periods 
increase  sfightly  for  a  number  of  years  and  then  decrease 
again,  showing  possibly  the  presence  of  a  third  body.  Some- 
times the  minima  as  determined  photographically  do  not 
occur  at  the  times  found  by  visual  observations. 

289.  Variable  Stars  of  the  Beta  Lyras  Type. — Variable 
stars  of  the  Beta  Lyrse  type  are  closely  related  to  those 
which  have  been  considered ;  in  fact,  the  distinction  between 
the  two  classes  seems  to  be  disappearing.  Their  fight  varies 
continuously  from  maximum  to  minimum  and  back  to  maxi- 
mum again.  The  maxima  are  afi  equal,  but  as  a  rule  there 
are  two  unequal  minima.  The  standard  star  of  this  class  is 
Beta  Lyrffl  (Fig.  176),  which  is  one  of  the  earfiest  known 
variables  and  gives  the  class  its  name. 

The  explanation  of  the  Beta  Lyrse  variables  is  that  they 
consist  of  two  stars  revolving  in  such  small  orbits  compared 


CH.  XIII,  290]       THE    SIDEREAL   XJNIVERSE 


519 


Z.5 

3.6 
Z.7 

/ 

N 

/ 

N 

1 

\ 

\ 

1 

\ 

/ 

^ 

Ly 

' 

\ 

J 

V 

DArS          1                  I                  3                  4                  s                 e                  7 

Fig.  176.- 


to'  their  dimensions  that  the  intervals  in  which  neither  ob- 
scures the  other  are  very  short.  While, this  explanation 
satisfies  the  phenomena  in  a  general  way,  there  are  many 
troubles  in  connection  with  the  details.  For  example,  about 
a  dozen  minor  variations  in  the  light  curve  of  Beta  Lyrse 
have  been  detected,  or  at  least  strongly  suspected.  Moreover, 
the  spectroscopic 
data  are  often 
puzzling.  But, 
on  the  whole, 
astronomers  are 
satisfied  that  the 
ecUpse  explana- 
tion, is  the  true 
one,  and  the  gap 
between  the  fight 
curves  of  Algol 
and  Beta  Lyrae  is 
gradually  being 
filled.  In  fact,  Shapley  includes  many  stars  of  the  Beta  Lyrse 
type  among  ecfipsing  variables  of  the  Algol  type. 

290.  Variable  Stars  of  the  Delta  Cephei  Type.  —  The  star 
Delta  Cephei  has  given  its  name  to  a  third  class  of  variables. 
In  these  stars  the  fight  curves  are  periodic  with  periods  rang- 
ing from  a  few  hours  to  45  days.  But  that  which  particu- 
larly characterizes  these  stars  is  that  they  increase  very 
rapidly  in  brightness  from  minimum  to  maximum,  and  then 
decfine  much  more  slowly  with  many  minor  irregularities 
modifying  the  gradual  diminution  in  brightness.  The  char- 
acteristics of  their  fight  curves  are  given  in  Fig.  177.  There 
are  a  few,  however,  known  as  the  Geminids  after  Alpha 
Geminorum,'  whose  fight  curves  are  nearly  symmetrical  with 
respect  to  their  maxima. 

The  explanation  of  the  Cepheid  variables  has  been  a  very 
puzzfing  problem.  Clearly  their  fight  changes  are  not  ordi- 
nary eclipse  phenomena,  but  their  spectral  lines  shift  periodi- 


-  Light  curve  of  a  variable  star  of  the 
Beta  Lyrse  type. 


520    AN   INTRODUCTION   TO  ASTKUJNUMy    lch.  xiii,  isau 


cally  with  the  periods  of  their  Hght  variations.  The  natural 
conclusion  has  been  that  they  are  spectroscopic  binaries  and 
that  the  changes  in  Ught  are  abnormal  echpse  phenomena. 
While  the  hght  changes  and  spectral  shifts  agree  in  period, 
they  absolutely  disagree  in  phase.  That  is,  interpreting 
the  spectroscopic  data  in  the  ordinary  way,  these  stars  are 
brightest  when  the  principal  stars  are  approaching  the 
observer  and  faintest  when  they  are  receding,  instead  of 
having  their  minima  when  they  are  eclipsed.  Evidently 
there  are  inconsistencies  in  the  interpretations,  and  it  is 
questionable  whether  echpses  have  anything  whatever  to  do 

with  the  hght  va- 
riations of  these 
stars.  A  number 
of  other  explana- 
tions have  been 
suggested,  the 
most  plausible  of 
which  is  that  the 
light  variations 
are  due  to  in- 
ternal oscilla- 
tions of  the  stars 
produced  per- 
haps by  collisions  with  masses  of  planetary  dimensions.  It 
has  been  found  that  very  moderate  oscillations  would  account 
for  the  variations  in  the  rates  of  radiation.  According  to 
this  hypothesis,  the  shifts  of  the  spectral  hues  are  produced 
partly  by  internal  motions  of  the  stars  and  partly  by  the 
effects  of  alterations  in  pressure  of  the  radiating  parts. 

291.  Variable  Stars  of  Long  Period.  —  A  majority  of 
variable  stars  belong  to  the  class  whose  periods  range  from 
50  to  several  hundred  days.  They  are  not  periodic  in  the 
strict  use  of  the  term  which  is  applicable  to  the  Algol  variables, 
yet  their  hght  varies  in  an  approximately  periodic  manner. 
But  the  intervals  between  maxima,  or  between  minima,  are 


e  to 
e.30 

?.50 

1 

\ 

1 

\ 

\^ 

\ 

< 

Z.70 

■^ 

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k 

\ 

l\ 

\ 

/ 

^ 

/■" 

DAYS        12                 3                  -4                  J                  6                  7 

Fig.  177.- 


-  Light  curve  of  a  variable  star  of  the 
Delta  Cephei  tjrpe. 


CH.  XIII,  291]       THE   SIDEREAL  UNIVERSE 


521 


subject  to  some  irregularities,  and  their  luminosities  at  cor- 
responding phases  are  by  no  means  always  the  same. 

The  best-known  star  of  this  class  is  Omicron  Ceti,  the 
first  known  variable.  It  has  been  observed  through  more 
than  300  of  its  cycles,  and  yet  it  has  not  been  found  possible 
to  formulate  any  law  describing  accurately  its  hght  varia- 
tions. Its  maxima  and  its  minima  are  subject  to  as  great 
irregularities  as  the  intervals  between  corresponding  phases. 
In  1779  WilUam  Herschel  saw  it  when  it  was  nearly, as  bright 
as  Aldebaran,  while  4  years  later  it  was  not  visible  even 
through  his  telescope.  This  means  that  it  was  at  least  10,000 
times  as  bright 
at  its  maximum 
as  at  that  par- 
ticular mini- 
mum. Ordinarily 
its  maximum  is 
much  below  that 
observed  by  Her- 
schel in  1779, 
and  its  minimum 
is  considerably 
above  the  limit 
of  visibiUty  with 
his  telescope.  Omicron  Ceti  was  called  Mira,  the  wonderful, 
and  300  years  of  observation  have  only  added  to  the  mysteries 
associated  with  its  pecuhar  behavior. 

The  general  characteristics  of  the  Ught  curves  of  variable 
stars  of  long  period  is  a  slow,  but  gradually  accelerated, 
increase  in  brightness  followed  by  a  much  more  gradual 
decline.  The  spectroscope  shows  marked  changes  in  their 
spectra,  but  no  evidence  of  their  being  spectroscopic 
binaries.  They  are  nearly  all  red  and  are  probably  of  not 
very  high  temperatures.  The  cause  of  their  variation 
seems  to  he  within  the  stars  themselves,  yet  it  is  difficult 
to  imagine  any  internal  disturbances  which  would  f)rdduce 


JO 

n 

/' 

\ 

s 

s 

s 

^J 

s  , 

/ 

V. 

/ 

>^ 

rSAFS       t                Z                3                4                s               e               7 

FiG.  178. 


-  Light  curve  of  variable  star  of  long 
penod. 


522    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  zm 

the  remarkable  fluctuations  which  are  observed  in  many- 
stars  of  this  class. 

292.  Irregular  Variable  Stars.  —  In  addition  to  the  classes 
of  variable  stars  so  far  enumerated,  there  are  others  whose 
variations  have  no  semblance  of  periodicity.  Some  flash 
out  with  relatively  great  brilliancy  after  intervals  usually 
counted  in  years.  These  stars  are  generally,  if  not  always, 
red.  Others  unaccountably-  fade  away  now  and  then  and 
sometimes  become  invisible  through  good  telescopes,  even 
though  they  had  been  ordinarily  visible  with  the  unaided  eye. 
These  stars  are  sometimes  associated,  at  least  apparently, 
with  faint  nebulous  masses. 

293.  Cluster  Variables.  —  A  very  interesting  and  im- 
portant discovery  was  made  in  the  last  decade  of  the  nine- 
teenth century  by  Bailey  at  the  South  American  branch  of 
the  Harvard  Observatory.  He  found  that  in  the  great 
globular  cluster,  Omega  Centauri,  125  stars  were  variable 
out  of  the  3000  which  he  examined.  He  and  other  astron- 
omers have  found  similar  variables  in  many  other  globular 
star  clusters.  In  a  given  cluster  the  range  of  variability  is 
nearly  the  same,  usually  a  magnitude  or  two,  the  character 
of  the  light  variation  is  essentially  the  same,  and  the  periods 
are  approximately  the  same,  generally  less  than  24  hours. 
Their  light  curves  are  closely  similar  to  those  of  the  variables 
of  the  Delta  Cephei  type,  and  it  is  really  a  question  whether 
the  cluster  variables  should  be  considered  a  separate  class. 
The  brightness  increases  with  great  rapidity  from  their 
minimum  to  a  luminosity  at  maximum  from  two  to  six  times 
as  great.  Then  they  diminish  in  brightness  much  more 
slowly  to  their  minimum,  at  which  they  remain  nearly 
stationary  for  a  few  hours  at  most. 

The  approximately  equal  periods  and  range  of  variation 
of  the  cluster  variables  indicate  that  they  are  very  much 
aUke  in  spite  of  the  enormous  distances  which  separate  them. 
Possibly  they  were  once  much  more  ahke  and  now  differ  to 
some  extent  because  of  slightly  different  courses  of  evolu- 


CH.  XIII,  294]       THE    SIDEREAL  UNIVERSE  623 

tion  or  present  environment.  Or,  possibly,  though  not 
probably,  there  is  some  great  common  cause  for  their  changes, 
a  force  causing  pulsations  in  scores  of  stars  distributed  widely 
throughout  the  clusters.  Although  nearly  2000  of  these 
objects  have  already  been  discovered  and  studied,  astrono- 
mers have  no  idea  as  to  the  reasons  for  their  peculiarities. 

294.  Temporary  Stars.  —  Occasionally  stars  have  been 
observed  to  blaze  forth  in  parts  of  the  sky  (mostly  in  the 
Milky  Way)  where  none  had  previously  been  seen,  and  then 
to  sink  away  into  obscurity  in  the  course  of  a  few  weeks  or 
months.  They  are  characterized  by  a  sudden  rise  to  one 
great  maximum  of  brilUancy  which,  notwithstanding  later 
temporary  increases,  is  never  repeated.  One  of  the  most 
remarkable  of  these  stars  of  which  there  are  any  records 
blazed  out  in  Cassiopeia  in  1572  and  was  for  a  time  as  bright 
as  Venus.  This  is  the  star  which  attracted  the  attention  of 
Tycho  Brahe  and  turned  him  to  astronomy.  The  interest  of 
Kepler  also  was  stimulated  by  the  discovery  of  a  temporary 
star  in  Ophiuchus  in  1604.  At  its  maximum  it  was  as  bril- 
liant as  Jupiter.  It  must  not  be  supposed  all  temporary 
stars  are  so  brilliant,  for  only  a  few  rise  to  such  splendor. 

In  recent  times  the  number  of  temporary  stars  discovered 
has  greatly  increased,  both  because  more  observers  are 
scanning  the  sky  than  eve?  before,  and  more  especially  be- 
cause they  are  now  recorded  by  photography.  In  the  last 
30  years  19  of  these  objects  have  been  discovered,  15  of 
which  were  found  first  on  the  photographic  record  of  the  sky 
which  is  being  secured  at  the  Hd.rvard  College  Observatory. 
Only  10  of  these  stars  were  discovered  from  1572  to  1886, 
when  the  photo^aphy  of  the  sky  was  first  systematically 
begun  at  Harvard. 

Temporary  stars  are  called  novce,  or  new  stars.  A  de- 
scription of  one  of  them  will  give  a  good  idea  of  the  charac- 
teristics of  all  of  them.  One  of  the  most  interesting  and  best 
studied  novae  of  recent  times  is  the  one  discovered  by  Ander- 
son, February  22,  1901,  in  Perseus.     On  the  23d  of  February 


524    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  294 


4t  was  brighter  than  Capella,  while  an  examination  of  the 
photographs  of  the  region  taken  by  Pickering  and  by  Stanley 
WiUiams  showed  that  on  the  19th  it  was  not  brighter  than 
the  12th  magnitude.  In  the  short  space  of  four  days  its 
rate  of  radiation  had  increased  more  than  20,000  fold. 
Twenty-four  hours  later  it  lost  one  third  of  its  hght,  and 
within  a  year  it  had  dwindled  to  the  12th  magnitude,  or  near 
the  limits  of  visibility  with  a  telescope  of  considerable  power. 
Its  light  curve  for  the  first  three  months  after  its  maximum 

is  shown  in  Fig. 


N 


Fig.   179.  —  Light  curve  of  Nova  Persei. 


179. 

The  changes  in 
the  spectra  of  the 
novse  are  as  re- 
markable as  their 
changes  in  lumi- 
nosity.  Very 
early  in  their  de- 
velopment they 
have  (at  least  in 
case  of  those 
stars  which  were  observed  early)  dark-line  spectra.  Shortly 
thereafter  bright  lines  appear.  In  the  case  of  Nova  Aurigse, 
discovered  in  1892,  and  the  first  temporary  star  whose  spec- 
trum was  examined  in  any  detail,  the  dark  lines  and  bright 
Hnes  were  both  visible  at  one  time.  The  displacement  of  the 
bright  lines  showed,  on  the  basis  of  the  Doppler-Fizeau 
principle,  a  velocity  away  from  the  earth  of  over  200  miles  per 
second,  while  the  dark  hnes  showed,  on  the  same  basis,  an 
approach  toward  the  earth  of  more  than  300  miles  per  second. 
There  are  abundant 'grounds  for  doubting  the  correctness  of 
this  interpretation,  but  no  satisfactory  explanation  is  at  hand. 
These  phenomena  are  characteristic  of  novse  in  general.  As 
they  become  fainter  the  dark  lines  vanish  and  the  bright  lines 
characteristic  of  nebulse  appear,  except  that  in  the  novse  they 
are  broad  while  they  are  narrow  in  the  nebulae. 


CH.  XIII,  294]       THE    SIDEREAL  UNIVERSE 


525 


The  most  interesting  thing  observed  in  connection  with 
Nova  Persei  was  the  nebulous  matter  which  was  later  found 
around  it.  Its  existence  was  first  shown  on  photographs  by- 
Wolf  taken  August  22  and  23,  1901.  Later  photographs  by 
Perrine  and  Ritchey  showed  that  it  was  gradually  becoming 
visible  at  increasing  distances  from  the  star.  It  looked  as 
though  the  star  had  ejected  luminous  matter,  but  it  was 
found  on  computation  that,  if  this  were  the  correct  explana- 
tion, the  expelled  matter  must  have  been  leaving  the  star 


Sy;i--y^S^^.'  >.'. 

.  •♦  .  •,:    ■  :. '■....      , .  •  ;•.;' . 

■  •.v'.v   •  ■  •      .  •'■."•■•■  ••^: .  ••  ■• 

■  -.:•..-■  ..  .■..".'■.  !.;•  :■-:.• ' , '■ . 

.  .    •  •« -^  -i -»  •.   ■;     .  •; .-  • 

..  ••'■•:.*.v.lV:..-.-;  i-'.  =■:"•■ 

Fig.  180.  —  Nebulosity  surrounding  Nova  Persei  on  Sept.  20  and  Nov.  13, 
1901.     Photographed  by  Ritchey  at  the  Yerkes  Observatory. 


with  stbout  the  velocity  of  light.     This,  of  course,  is  improb- 
able if  not  impossible. 

The  temporary  stars  demand  explanation.  The  theory 
was  suggested  by  Kapteyn  and  W.  E.  Wilson,  and  expounded 
in  detail  by  Seeliger,  that  there  is  invisible  nebulous  or 
meteoric  matter  lying  in  various  parts  of  space,  particu- 
larly in  the  region  occupied  by  the  Milky  Way  (there  is  con- 
firmatory evidence  of  this  hypothesis) ;  that  there  are  dark 
or  very  faint  stars  (confirmed  by  phenomena  of  ecHpse 
variables) ;  that  the  dark  stars,  rushing  through  the  nebulse, 
blaze  into  incandescence  as  meteors  glow  when  they  enter 
the  earth's  atmosphere ;  that  the  heating  is  only  superficial 
and  quickly  dies  away,  to  be  partially  revived  once  or  twice 
by  encounters  of  the  stars  Avith  stray  nebulous  wisps ;   and 


526    AN   INTRODUCTION   TO  ASTRONOMY    ten.  xm,  ^»t 

that  the  nebulous  ring  observed  around  Nova  Persei  became 
visible  as  it  was  illuminated  by  the  light  from  the  star  itself. 

The  explanation  of  Kapteyn  at  first  seems  plausible,  but 
there  are  serious  objections  to  it.  In  the  first  place,  the 
photographs  of  Nova  Persei  indicate  strongly  that  the  ex- 
panding nebulous  ring  surrounding  it  was  due  to  something 
actually  moving  out  radially  from  the  star.  In  the  second 
place,  the  density  of  the  nebula  demanded  to  account  for 
the  enormous  rise  in  luminosity  is  impossibly  high.  In  the 
third  place,  the  fact  that  the  star  stays  at  its  maximum  only 
a  very  short  time  implies  a  nebula  whose  thickness  is  in- 
credibly small. 

Lindemann  has  developed  the  hypothesis  that  novse  are 
produced  by  collisions  of  stars  with  stars.  If  one  star  should 
encounter  another  in  central  collision  with  the  great  speed 
at  which  they  would  move  as  a  consequence  of  their  initial 
motion  and  mutual  gravitation,  the  heat  generated  would 
be  enormous.  If  they  were  of  equal  mass  and  started  from 
rest,  the  heat  developed  would  be  five  sixths  of  that 
which  would  be  generated,  according  to  the  principles  of 
Helmholtz,  by  the  contraction  of  both  of  them  from  infinite 
expansion.  This  heat  would  be  developed  in  a  few  hours, 
or  days  at  the  most,  and  the  temperature  of  the  combined 
mass  would  rise  enormously.  But  with  increase  of  tem- 
perature there  would  be  corresponding  expansion,  which 
would  result  in  a  diminution  of  the  temperature.  If  the 
stars  were  originally  gaseous,  the  final  temperature  after 
expansion  would  be  lower  than  that  before  collision  because 
the  conditions  are  the  opposite  of  those  in  Lane's  law  (Art. 
216),  according  to  which  the  temperature  of  a  gaseous  star 
increases  as  it  loses  heat  by  radiation  and  contracts.  Or, 
stated  directly^  if  heat  could  be  applied  to  a  gaseous  star  by 
radiation  or  otherwise,  it  would  expand  and .  increase  its 
potential  energy  at  the  expense,  not  only  of  all  the  heat  sup- 
plied, but  also  partly  at  the  expense  of  that  which  it  already 
possessed. 


CH.  xni,  295]       THE   SIDEREAL  UNIVERSE  527   s 

While  in  a  general  way  the  collision  theory  of  the  origin 
of  novae  corresponds  with  the  observations,  it  is  not  without 
difficulties.  Obviously,  actual  collisions  of  stars  would  be 
excessively  rare  phenomena.  Lindemann  finds  that  in 
order  to  accoiunt  for  the  observed  number  of  temporary 
stars  there  must  be  about  4000  times  as  many  dark  stars  as 
there  are  bright  ones.  Such  a  large  number  of  obscure 
masses  would  radically  modify  the  dynamics  of  the  stellar 
system  (Art.  279) ;  and  it  is  generally  regarded  as  improb- 
able that  so  many  of  them  exist. 

295.  The  Spectra  of  the  Stars.  —  The  spectra  of  the  stars 
differ  as  greatly  as  their  colors.  They  were  first  classified 
in  1863,  by  Secchi,  who  divided  them  into  four  groups. 


Fig.   181.  —  The  spectrum  of  Sirius  ^Secchi's  I'ype  1, 


While  more  powerful  instruments  have  shown  many  new 
facts  and  have  made  it  necessary  to  add  many  new  sub- 
classes, the  four  types  described  by  Secchi  still  form  a  general 
basis  for  classification.  A  more  detailed  classification,  which 
is  now  much  used,  was  devised  by  E.  C.  Pickering,  Miss 
Maury,  Mrs.  Fleming,  and  Miss  Cannon  in  connection  with 
the  great  photographic  survey  of  stellar  spectra  which  is 
being  made  at  the  Harvard  College  Observatory. 

Type  I.  Stars  of  Secchi's  first  type  are  blue  or  bluish 
white.  Examples  are  Sirius,  Vega,  and  all  bright  stars  in 
the  Big  Dipper  except  the  first  one.  Nearly  half  of  all  stars 
examined  are  of  this  type.  Their  spectra  are  brightest 
toward  the  violet  end,  indicating  presumably  that  they  are 
at  high  temperatures.  The  spectrum  of  Sirius  is  shown  in 
Fig.  181. 


528    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  295 


Type  I,  in  Secchi's  system,  includes  Types  B  and  A  of 
the  Harvard  system.  Type  B  is  often  called  the  Orion  type 
because  of  the  abundance  of  these  stars  in  Orion,  or  the 
helium  type,  because  the  absorption  lines  are  due  almost 
entirely  to  helium,  while  the  metallic  lines  which  are  char- 
acteristic of  the  sun's  spectrum  are  absent.  The  Type  A, 
or  Sirian  stars,  are  characterized  by  strong  hydrogen  absorp- 
tion lines  in  their  spectra,  and  almost  complete  absence  of 
metalhc  lines. 

Type  II.  The  stars  of  the  second  type  are  somewhat 
yellowish ;  they  are  called  solar  stars  because  their  spectra  are 


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i 

■ 

Fig.  182.  —  Spectrum  of  Beta  Geminorum  (Harvard  Class  K).   Photographed 
at  the  Yerkes  Observatory. 

similar  to  that  of  the  sun.  That  is,  the  lines  of  helium  are 
absent,,  the  hues  of  hydrogen  are  still  present,  and  there 
are  many  fine  metallic  lines.  The  stars  of  the  second  type 
are  about  as  numerous  as  those  of  the  first  type. 

Secchi's  second  type  includes  three  classes  of  the  Harvard 
system.  Those  nearest  hke  the  Sirian  stars  are  called  Type 
F,  or  the  calcium  type.  In  their  spectra  the  hydrogen  lines 
are  still  conspicuous,  though  somewhat  reduced  in  density, 
and  two  lines,  known  as  H  and  K,  due  to  calcium  have 
become  conspicuous.  Following  the  class  F  is  the  class  G, 
of  which  the  sun  is  a  typical  member.  Then  come  the  stars 
of  Type  K,  of  which  Beta  Geminorum  and  Arcturus  are  ex- 
amples, in  which  the  intensity  of  the  hydrogen  lines  is  re- 
duced until  they  are  less  conspicuous  than  some  of  the 


CH.  XIII,  isaoj        T±1E    81DKKEAL  UNIVERSE 


529 


metallic  lines.    The  spectra  of  these  stars  are  given  in  Figs. 
182  and  183. 

Type  III.  Stars  of  the  third  type  are  red,  and  the  two 
most  conspicuous  examples  of  them  are  Antares  and  Betel- 
geuze.  Only  about  500  of  these  stars  are  known,  and  many 
of  them  are  variable.  Their  spectra  show  heavy  absorption 
bands,  due  almost  entirely  to  titanium  oxide,  which  are 
sharp  on  their  borders  toward  the  violet  and  which  gradually 
fade  away  toward  the  red.  The  fact  that  a  compound  exists 
in  these  stars  indicates  that  their  temperatures  are  lower 
than  those  of  Types  I  and  II.  The  same  thing  is  indicated 
by  their  colors  in  accordance  with  the  first  law  of  spectrum 
analysis  (Art.  223).  In  all  known  cases  they  have  very  small 
proper  motions,  which  means  that  they  are  immensely  re- 


FiG.  183.- 


-  Spectrum  of  Arcturus  (Harvard  Class  K) . 
Yerkes  Observatory. 


Photographed  at  the 


mote  from  the  sun.  Hence  such  brilliant  stars  as  Antares 
and  Betelgeuze,  whose  light  is  largely  absorbed,  must  be 
enormous  objects.  They  are  almost  certainly  many  thou- 
sand times  greater  in  volume  than  our  own  sun. 

The  stars  of  Secchi's  third  type  are  of  Type  M  in  the 
Harvard  system.  They  are  divided  into  two  chief  sub- 
classes, Ma  and  Mb ;  a  third  subclass  Md  includes  the  long- 
period  variable  stars  whose  spectra  show  bright  hydrogen 
lines  in  addition  to  the  bands  characteristic  of  the  whole  type. 

Type  IV.  The  250  stars  of  Secchi's  fourth  type  are  all 
faint  and  of  a  deep  red  color.  Their  spectra  have  heavy 
absorption  bands,  or  flutings,  sharp  on  the  red  side  and  in- 
definite on  the  violet,  being  in  this  respect  opposite  to  the 
stars  of  the  third  type.  The  absorption  bands  in  this  case 
are  probably  due  to  carbon  compounds.  These  stars  are  all 
2m 


530    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  295 

very  remote  from  the  sun,  and  nothing  is  known  of  their 
absolute  magnitudes,  or  of  their  masses  and  dimensions. 

The  Wolf-Rayet  Stars.  There  is  another  class  of  stars, 
discovered  in  1867  by  Wolf  and  Rayet  at  the  Paris  Observa- 
tory. They  are  Type  O,  having  five  subdivisions,  in  the 
Harvard  system.  Their  spectra  consist  of  fairly  contin^ious 
backgrounds  on  which  are  superimposed  many  dark  hnes 
and  bands,  some  few  of  which  are  due  to  helium  and  hydro- 
gen, but  most  of  them  to  unknown  substances.  They  con- 
tain in  addition  many  bright  lines.  The  metallic  lines  of  the 
solar  spectrum  are  quite  unknown  in  these  stars.  Of  the 
more  than  100  stars  of  this  type  so  far  discovered,  all  are 
situated  either  in  the  Milky  Way  or  in  the  Magellanic  Clouds 
in  the  southern  heavens,  which  have  most  of  the  characteris- 
tics of  the  Milky  Way. 

296.  Phenomena  Associated  with  Spectral  Types.  —  A 
large  number  of  phenomena  combine  to  show  that  the  classi- 
fication of  stars  according  to  their  spectra  is  on  a  funda- 
mental basis.  The  order  of  arrangement  from  the  simplest 
to  the  most  complex  spectra  is  : 

Secchi's  Types :     Wolf-Rayet ;  I ;  II ;  III ;      IV. 

Harvard  Types :  O ;  B,  A ;       F,  G,  K ;       M ;       N. 

If  the  gaseous  nebulae  were  included,  they  would  be  put 
•ahead  of  the  Wolf-Rayet  stars.  There  is  a  fairly  continu- 
ous sequence  of  spectra  from  Type  O  to  Type  M,  but  there 
is  an  abrupt  break  between  Types  M  and  N, 

The  principal  phenomena  which  are  associated  with  the 
spectral  types  and  which  agree  on  the  whole,  in  arranging 
the  stars  in  the  same  order,  are  : 

(a)  The  average  radial  velocities  of  the  stars,  determined 
largely  at  the  Lick  Observatory  and  its  southern  branch, 
and  discussed  by  Campbell,  are  slowest  for  stars  of  Type  B 
and  increase  to  Type  M.  The  results,  as  given  by  Campbell, 
with  velocities  expressed  in  miles  per  second,  are  : 

Types :  B,      A,      P,      G,       K,       M,      Planetary  Nebulae. 

Velocities:    4.0,    6.8,    8.9,    9.3,    10.4,    10.6,  15.7 


uH.  XIII,  z»Dj        ijtiiji   biJJJiKi*:AL  UNIVERSE  531 

"^  (6)  The  average  velocities  of  the  stars  across  the  line  of 
sight,  as  determined  by  Lewis  Boss,  show  a  similar  relation 
to  the  spectral  type.    The  results  are : 

Types:  B,  A,  F,  G,  K,  M. 

Velocities:         3.9,         6.3,         10.0,         11.5,        9.4,         10.6. 

These  results  together  with  those  depending  on  the  spectro- 
scope estabhsh  the  fact  that  the  stars  of  Types  B  and  A 
move  on  the  average  only  about  half  as  fast  as  those  of 
Types  G,  K,  and  M. 

(c)-  In  Kapteyn's  star-stream  I,  the  B  and  A  stars  are 
relatively  numerous,  the  F,  G,  and  K  stars  occur  less  fre- 
quently, and  the  red  stars  are  very  few  in  number.  In  the 
star-stream  II,  the  B  and  A  stars  are  not  numerous,  the  F, 
G,  and  K  stars  occur  in  relatively  great  numbers,  and  the 
M  stars  are  scarce. 

(d)  While  there  are  two  great  star-streams,  there  are  very 
many  divergencies  from  them  on  the  part  of  individual 
stars.  The  stars  of  Type  B  scarcely  show  the  star-stream- 
ing tendency,  those  of  Type  A  conform  very  closely  to  the 
two  streams,  and  succeeding  types  show  more  and  more  of 
heterogeneity  of  motion. 

(e)  On  considering  only  stars  brighter  than  magnitude  6.5 
so  as  not  to  have  the  results  influenced  by  the  myriads  of 
remote  stars,  it  is  found  that  the  B  stars  are  10  times  as 
numerous  in  the  Milky  Way  as  near  its  poles,  the  A  stars 
are  less  strongly  condensed  in  the  Milky  Way,  and  finally, 
after  continuous  gradation  through  the  various  types,  the 
M  stars  are  scattered  uniformly  over  the  sky. 

(f)  For  a  given  magnitude  the  stars  of  Type  B  are  more 
remote  than  those  of  Type  A,  which,  in  turn,  are  more  re- 
mote than  those  succeeding  down  to  Type  G ;  then,  beyond 
Type  G,  the  distances  increase  to  stars  of  Type  M,  whose 
distances  are  exceeded  only  by  the  B  stars.  This  means, 
of  course,  that- the  B  stars  are  most  luminous,  the  A  stars 
less  luminous,  the  G  stars  least  luminous,  while  the  M  stars 
are  more  luminous  than  any  except  the  B  stars. 


532    AN   INTRODUCTION  TO  ASTRONOMY   [ch.  xiii,  296 

(g)  The  proportion  of  B  stars  which  are  spectroscopic 
binaries  is  large,  the  proportion  is  less  for  the  A  stars  and 
it  decreases  through  the  list  of  types  to  M. 

(h)  Lower  limits  to  the  combined  masses  of  spectj-oscopic 
binaries  can  be  determined  (Art.  285).  The  average  mass 
of  those  of  Type  B  is  about  7.5  times  the  average  mass  of 
all  other  types. 

(i)  The  average  period  of  spectroscopic  binaries  of  Type  B 
is  very  short,  the  average  is  a  little  longer  for  stars  of  Type  A, 
and  increases  through  Types  F,  G,  K,  and  M. 

(j)  The  average  eccentricity  of  the  orbits  of  spectroscopic 
binaries  is  small  for  stars  of  Type  B,  is  larger  for  stars  of 
Type  A,  and  is  increasingly  larger  for  star^  of  the  Types  F, 
G,  and  K,  in  order. 

297.  Evolution  of  the  Stars.  —  All  the  resources  of  science 
have  been  taxed  to  the  utmost  in  attempting  to  discover  the 
present  constitution  and  properties  of  the  sidereal  system. 
At  the  best,  astronomers  have  barely  begun  to  explore  the 
wonders  of  that  part  of  infinite  space  which  is  within  the 
reach  of  modern  instruments.  Moreover,  their  observational 
experience  is  limited  to  a  moment  of  time  compared  with 
the  immense  ages  required  for  appreciable  changes  to  take 
place  in  the  heavenly  bodies.  Hence  it  may  seem  presump- 
tuous for  them  to  attempt  to  discover  the  mode,  or  modes, 
of  evolution  of  the  stars.  Any  theories  of  stellar  evolution 
that  may  be  developed  at  the  present  time  are  probably  no 
more  than  first  approximations,  and  they  may  be  entirely 
wrong. 

Astronomers  almost  universally  hold  that  the  stars  have 
contracted  from  the  nebulae,  and  most  of  them  believe  that 
with  increasing  age  they  have  gone,  or  are  now  going,  suc- 
cessively and  in  order  through  the  spectral  types  B,  A,  F, 
G,  K,  and  M.  The  B  stars  are  of  very  high  temperature 
and  are  pouring  out  radiant  energy  at  an  extravagant  rate. 
After  they  cool  somewhat  it  is  supposed  that  they  become 
stars  of  Type  A.     Their  spectra  are  supposed  to  be  simple 


CH.  XIII,  297]       THE    SIDEREAL  UNIVERSE  533 

because  all  compounds,  and  possibly  some  elements,  are 
broken  up  and  dissociated  at  those  high  temperatures.  With 
further  loss  of  hekt  they  are  supposed  to  pass  successively 
through  the  other  spectral  types  until,  at  the  M  stage,  com- 
pounds exist  in  their  atmospheres.  Beyond  the  M  stage  their 
light  diminishes  and  they  finally  become,  in  the  course  of 
time,  cold  and  dark,  and  they  remain  in  this  condition  until, 
perhaps,  they  are  again  reduced  to  the  nebulous  state  by 
collision  with  other  stars.  All  the  forms  in  the  chain  from 
nebulae  to  relatively  dark  stars  are  known  to  exist  -from 
observational  evidence.  The  many  other  characteristics 
which  arrange  the  stars  in  nearly,  or  exactly,  the  same  order 
are  regarded  as  strongly  supporting  the  theory. 

The  theory  of  the  evolution  of  the  stars  has  strong  resem- 
blances to  the  Laplacian  theory  of  the  development  of  the 
solar  system.  This  is  only  natural  in  view  of  the  general 
acceptance  of  the  theory  of  Laplace  almost  up  to  the  present 
time.  As  additional  facts  have  been  discovered  they  havp 
been  placed  in  this  scheme,  often  without  inquiring  if  they 
would  not  fit  as  well  in  some  other  theory. 

Laplace  started  with  an  intensely  heated  and  widely  ex- 
panded solar  nebula  and  he  supposed  that  it  has.  cooled 
down  to  its  present  tempjerature.  Helmholtz  supplemented 
and  corrected  this  theory  by  proving  that  contraction  would 
develop  an  enormous  amount  of  heat  and  greatly  retard  the 
process  of  cooling.  The  conclusions  of  Helmholtz  have  been 
given  place  in  the  theory  of  the  evolution  of  the  stars.  Lane 
made  a  further  very  .important  supplement  to  the  work  of 
Laplace  -When  he  proved  that  if  a  body  in  a  monatomic  gas- 
eous state  contracts,  heat  is  produced  in  quantities  not  only 
sufficient  to  make  up  for  that  which  had  been  radiated  away, 
but  also  sufficient  actually  to  increase  its  temperature.  In 
spite  of  the  fact  that  the  results  of  Lane  have  been  current 
for  almost  fifty  years,  they  have  often  been  ignored  in  their 
application  to  the  evolution  of  the  stars.  If  the  stars  of 
any  type  are  in  a  tenuous  monatomic  gaseous  condition  and 


534    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  297 

contract,  their  temperature  will  inevitably  rise  and  continue 
to  rise  until  they  cease  to  be  entirely  gaseous  and  monatomic. 

Consequently;  if  the  stars  of  the  types  B,  A,  F,  G,  K,  M 
are  in  the  order  of  decreasing  temperature  and  are  gaseous, 
the  logical  conclusion  on  the  basis  of  the  supplements  to 
Laplace's  theory  is  that  the  evolution  proceeded  in  the  re- 
verse order.  Of  covu-se,  the  stars  may  not  all  be  completely 
gaseous.  This  has  given  rise  to  the  theory,  proposed  by 
Lockyer  and  amplified  and  ably  supported  by  Russell,  that 
the  nebulse  contract  into  tenuous  red  stars  of  Type  M  which 
have  low  temperatm-es ;  with  loss  of  heat  they  contract, 
their  temperatures  rise,  their  spectra  become  simpler  vmtil 
they  reach  their  cUmax  in  Types  A  and  B ;  after  this  they 
cease  to  be  completely  gaseous,  and  with  increasing  conden- 
sation and  liquefaction,  their  temperatiu-es  decline  and  their 
spectra  proceed  back  through  the  types  F,  G,  and  K  to  M. 
The  cogency  of  the  arguments  on  which  these  conclusions 
rest  cannot  be  denied,  and  many  observational  data  are 
quite  in  harmony  with  them.  But  there  are  also  some  things 
(for  example,  the  high  velocities  of  the  nebulse.  Art.  301) 
which  have  been  thought  to  be  strongly  opposed  to  them. 
The  two  theories  are  alike  in  starting  from  nebulse  and  end- 
ing with  cold  and  hfeless  suns. 

298.  The  Tacit  Assumptions  of  the  Theories  of  Stellar 
Evolution.  — ■  In  every  theory  there  are  many  more  or  less 
tacit  assmnptions,  some  of  which  may  be  of  great  impor- 
tance. It  has  been  found  by  a  large  amount  of  experience  that 
errors  more  frequently  enter  through  unexpressed  hypotheses 
than  in  any  other  way.  This  has  been  particularly  true  in 
mathematics  where  it  is  relatively  easy  to  determine  pre- 
cisely the  location  of  the  error  that  has  been  made  in  any 
course  of  reasoning.  It  follows  that  one  of  the  best  ways  al 
avoiding  errors  is  to  express  fully  all  the  hypotheses  on 
which  reasoning  is  based.  And  quite  aside  from  this,  it  is 
useful  and  important  to  know  all  the  bases  on  which  con- 
clusions actually  rest.    Consequently,  the  tacit  and  imper- 


CH.  xiii,  298]       THE    SIDEREAL  UNIVERSE  535 

fectly  established  assumptions  on  which  the  present  theories 
of  stellar  evolution  are  founded  will  be  enumerated ;  it  will 
be  foimd  that  at  the  present  time  most  of  them  must  remain 
simply  assumptions. 

(a)  It  is  assumed  that  the  evolution  of  the  stars  is  from  neb- 
uloe  to  dense  bodies  and  not  in  the  opposite  direction. 

The  best  evidence  in  support  of  or  against  a  proposition 
is  usually  observational;  when  observational  evidence  is 
lacking,  we  must  resort  to  reasoning  based  as  far  as  possible 
on  principles  which  have  been  established  by  experience. 

There  is  as  yet  no  observational  evidence  that  nebulae  or 
stars  contract ;  observations  have  extended  over  so  short  a 
time  that  it  could  not  be  expected.  On  the  other  hand,  in 
the  case  of  the  novae,  stars  are  observed  to  acqmre  the  char- 
acteristics of  the  Wolf-Rayet  stars,  which  border  on  the 
planetary  nebulae.  Of  course,  this  may  be  quite  excep- 
tional, but  it  should  not  be  neglected.  Consequently,  in  this 
matter' there  is  no  conclusive  observational  evidence. 

The  principal  known  force  wljich  tends  to  produce  con- 
densation is  gravitation.  In  the  case  of  the  stars  this  force 
is  balanced  by  the  expansive  forces  due  to  their  high  tem- 
peratures. If  their  heat  is  produced  only  by  their  con- 
traction, as  the^  lose  heat  by  radiation,  they  certainly  con- 
tract. But  the  contraction  theory  is  inadequate  to  explain 
the  heat  which  the  sun  has  radiated  (Art.  219),  and  it  seems 
very  probable,  if  not  altogether  certain,  that  stars  have 
other  important  sources  of  energy.  As  has  been  suggested, 
the  heat  of  the  pun  is  probably  due  in  part  to  the  disinte- 
gration of  radioactive  substances.  Perhaps  in  the  extreme 
conditions  of  pressure  and  temperature  prevailing  in  the 
deep  interiors  of  stars  the  process  of  disintegration  is  greatly, 
accelerated  and  is  going  on  in  all  elements.  And  probably 
there  are  very  important  sources  of  energy  not  now  sus- 
pected, just  as  the  sub-atomic  energies  were  not  suspected 
a  few  years  ago. 

Now  suppose  the  amount  of  energy  generated  in  a  star 


536    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  298 

in  all  these  ways  is  greater  than  that  radiated.  Then  the 
star  will  inevitably  expand  and  its  temperature  will  fall, 
because  with  increased  dim,ensions  gravitation  cannot  bal- 
ance so  high  a  temperature.  If  the  process  continues,  the 
star  will  expand  to  a  nebula,  which  will  necessarily  have  a 
low  temperature.  In  this  case  the  direction  of  evolution 
would  be  reversed.  But  as  the  star  expands,  the  conditions 
in  its  interior  are  changed,  and  the  production  of  energy 
might  be  reduced  so  that  it  would  only  equal  that  radiated. 
-In  this  case  the  star  would  reach  a  condition  of  equihbrium 
which  would  be  indefinitely  maintained  unless  the  sub- 
atomic and  other  possible  sources  of  energy  were  ultimately 
exhausted,  and  it  seems  certain  that  they  would  become  ex- 
hausted. Then  the  star  would  contract  if  its  disintegrated 
products  still  obeyed  the  law  of  gravitation,  and  its  evolu- 
tion would  proceed  in  the  direction  assumed  in  current 
theories,  though  at  a  greatly  retarded  rate. 

In  reaching  the  conclusions  which  have  been  set  forth  it 
has  been  assumed  that  the  masses  of  the  stars  are  constant. 
It  is  clear  that  their  masses  probably  are  increased  somewhat 
by  the  accretion  of  meteoric  matter  and  individual  mole- 
cules, but,  so  far  as  may  be  judged  from  the  sun,  this  is  not 
an  important  factor.  It  is  quite  certain  that  the  sun  is 
emitting  electrified  particles  in  great  mmabers  and  with  high 
velocities.  Probably  the  auroral  displays  in  the  earth's  at- 
mosphere are  produced  by  such  particles  impinging  on  the 
molecules  in  the  tenuous  gases  at  great  altitudes.  In  view 
of  the  considerable  hght  sometimes  emitted  by  aurorse  and 
the  earth's  immense  distance  from  the  sun,  it  seems  prob- 
able that  the  sun  loses  these  particles  at  a  rate  which  makes 
the  process  important.  If  so,  the  stars  may  possibly  be  dis- 
integrating into  nebulffi.  For  example,  the  nebulosities 
around  the  Pleiades  (Fig.  184)  may  have  come  out  from  these 
stars  instead  of  being  gradually  drawn  in  upon  them.  Be- 
sides this,  comets  give  numerous  examples  of  matter  being 
dispersed  in  space. 


CH.  XIII,  -Jasj       TMJi;    81DERBAL  UNIVERSE 


537 


Fig.  184.  —  The  Pleiades.  These  stars  are  surrounded  by  nebulous  masses 
of  enormous  volume.  Photographed  by  Bitchey  with  the  two-foot  reflector 
of  the  Yerkes  Observatory. 


538    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  298 

It  is  obvious  that  we  do  not  know  with  any  high  degree 
of  certainty  in  which  direction  stellar  evolution  is  proceed- 
ing. Sound  scientific  method  calls  for  keeping  both  of  them 
in  mind  until  a  decision  is  reached  on  the  basis  of  unequiv- 
ocal evidence.  Whichever  of  the  two  conclusions  may  pre- 
vail, the  result  will  be  unsatisfactory,  for  it  will  indicate  a 
universe  evolving  always  in  one  direction,  leaving  the  origin 
unexplained.  Possibly  there  are  changes  in  both  directions, 
and  it  may  be  that  stellar  evolution  in  some  way  and  on  a 
stupendous  scale  is  approximately  cyclical  Uke  most  of  the 
changes  which  come  entirely  within  the  range  of  our  experi- 
ence. 

(6)  It  is  assumed  that  all  stars  have  approximately  the  same 
chemical  constitution;  or,  if  not,  that  their  spectra  do  not  de- 
pend to  an  important  extent  upon  their  chemical  constitutions. 
One  or  the  other  of  these  assumptions  is  made  tacitly  when 
it  is  supposed  that  all  stars  pass  in  one  direction  or  the  other 
through  several  identical  spectral  types. 

The  spectroscope  proves  that  the  stars  contain  famihar 
elements;  it  does  not  prove  that  they  do  not  contain  some 
unknown  elements,  or  that  the  known  elements  occur  in  all 
stars  in  the  same  proportions.  The  great  diversities  on  the 
earth  make  it  natural  to  conclude  that  there  are  important 
differences  in  the  miUions  of  stars  in  the  heavens.  More- 
over, the  different  dimensions,  densities,  and  absorption 
spectra  of  the  planets  lead  to  the  same  conclusion.  The 
hypothesis  that  the  stars  are  of  approximately  identical  con- 
stitution must  be  considered  improbable  until  it  is  supported 
by  observational  evidence. 

It  is  too  bold  to  assume  that  if  the  stars  are  differently 
constituted  they  nevertheless  have  the  same  spectra  at  the 
same  temperatm-es.  But  the  assumption  actually  made  is 
not  quite  so  bad  as  it  at  first  seems,  for  the  stellar  spectra 
from  B  to  F,  and  even  G,  are  classified  primarily  on  the 
basis  of  their  hydrogen  emission  and  absorption  lines. 
Within  these  classes  there  is  opportunity  for  great  variety. 


CH.  XIII,  298]       THE    SIDEREAL  UNIVERSE  539 

and -indeed  variety  is  not  wanting.  There  is  nothing  ob- 
viously unsound  in  supposing  that  the  character  of  the  hydro- 
gen spectra  of  the  stars  depends  upon  their  temperatures. 
But  the  question  is  whether  a  star  which  has  only  hehum 
and  hydrogen  hues  can  ever  show  the  strong  metalUc  absorp- 
tion lines  which  are  characteristic  of  stars  of  Types  F  and  G. 
Fortunately,  there  is  now  direct  evidence  on  this  point,  for 
there  are  certain  variable  stars  which,  at  their  maxima, 
are  of  spectral  Types  B  or  A,  while,  at  their  minima,  they 
are  of  Types  F  or  G.  There  is  nothing  inherently  improb- 
able in  ascribing  these  changes  in  luminosity  and  spectra  to 


Fig.   185.  —  For  a  given  density,  the  more  massive  the  star  the  higher  its 
temperature. 

changes  in  temperature,  produced,  perhaps,  by  contracting 
and  expanding  oscillations  of  these  stars. 

{c)  It  is  assumed  that,  aside  from  the  rate  of  change,  the  evo- 
lution of  a  star  does  not  depend  on  its  mass.  In  considering 
this  point  the  assumption  that  the  spectrum  of  a  star  depends 
upon  the  temperature  of  its  radiating  surface,  or  radiating 
layer,  should  constantly  be  borne  in  mind. 

It  should  be  recalled  in  the  first  place  that  the  known 
masses  of  the  stars  differ  considerably  (Art.  284),  and  it  is 
improbable  that  the  few  which  are  known  cover  anjrwhere 
nearly  the  whole  range.  Consider  two  stars,  S  and  S',  Fig. 
185,  of  the  same  material  and  equal  density  but  one  having 
twice  the  mass  of  the  other,  and  fasten  attention  on  unit 


540    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  298 

volumes  at  any  corresponding  points  P  and  P'  in  their  in- 
teriors. The  pressure  on  the  unit  volume  at  P  is  greater 
than  that  on  the  unit  volume  at  P',  both  because  the  column 
PA  is  longer  than  P'A'  and  also  because  each  unit  mass  in 
PA  is  subject  to  a  greater  attraction  than  that  to  which  the 
corresponding  mass  in  P'A'  is  subject.  To  balance  the 
higher  pressure  in  the  larger  star  the  gaseous  mass  at  P 
must  have  a  higher  temperature  than  that  at  P'.  Conse- 
quently, if  two  stars  of  the  same  material  are  of  the  same 
density  at  corresponding  parts  and  are  of  unequal  masses, 
the  temperature  of  the  larger  star  at  all  points  from  its  center 
to  its  surface  is  higher  than  that  of  the  smaller  star ;  and  if 
the  spectrum  of  a  star  depends  primarily  on  its  temperature, 
their  spectra  are  different. 

A  mathematical  discussion  shows  that  if  two  stars  are  of 
the  same  material  and  of  equal  densities  at  corresponding 
points,  their  absolute  temperatures  are  as  the  squares  of 
their  radii.  On  combining  this  result  with  Lane's  law  that 
the  absolute  temperature  of  a  monatomic  gaseous  star  is 
inversely  as  its  radius,  it  is  found  that  the  absolute  temper- 
atures of  stars  of  equal  volumes  and  the  same  material  are 
proportional  to  their  masses. 

The  results  which  have  just  been  reached  are  very  im- 
portant, even  if  they  represent  the  physical  facts  only  ap- 
proximately, and  they  should  not  be  ignored  in  discussions 
of  stellar  evolution.  For  the  purposes  of  numerical  illustra- 
tion suppose  the  sun  is  gaseous  and  consider  a  star  of  the 
same  material  and  density  having  a  radius  twice  as  great. 
Its  mass  is  eight  times  that  of  the  sun.  By  the  first  law,  its 
temperature  is  four  times  that  of  the  sun.  Since  the  rate 
of  radiation  is  proportional  to  the  fourth  power  of  the  abso- 
lute temperature,  its  radiation  per  unit  area  is  256  times 
that  of  the  sun.  Since  its  radius  is  twice  that  of  the  sun, 
its  surface  is  4  times  greater,  and  its  whole  radiation,  or 
luminosity,  is  4  X  256  =  1024  times  that  of  the  sun.  That 
is,  two  stars  of  the  same  material  and  density,  whose  masses 


CH.  XIII,  298]       THE    SIDEREAL  UNIVERSE  541 

are  in  the  ratio  of  only  8  to  1,  differ  in  luminosity  in  the  ratio 
of  1024  to  1.  If  a  star  were  eight  times  more  massive  than 
the  sun,  it  would  have  a  spectrum  of  Type  B  or  A,  if  these 
spectra  indicate  high  temperatures,  and  it  would  be  a  star 
comparable  to  the  most  brilliant  ones  found  in  the  heavens. 
On  the  other  hand,  if  it  were  one  eighth  as  massive  as  the 
sun,  it  would  have  a  spectrum  characteristic  of  low  temper- 
atures (Type  M?),  and  would  be  a  feebly  luminous  body. 

Of  course,  it  is  not  necessary  that  other  stars  should  have 
the  same  density  as  the  sun.  It  is  known  from  eclipsing 
variables  that  comparatively  few  are  as  dense  as  the  sun, 
and  that  the  densities  may  be  as  small  as  one  hundredth  or 
even  one  thousandth  of  that  of  the  sun.  It  can  be  shown 
that  the  temperature  of  a  gaseous  star  is  proportional  to  the 
cube  root  of  the  product  of  the  square  of  the  mass  and  the 
density.  Hence,  in  order  that  a  star  having  a  density  one 
hundredth  that  of  the  sun  should  be  as  hot  as  the  sun,  its 
mass  must  be  about  10  times  greater.  But  under  these 
conditions  its  surface  and  luminosity  would  both  be  about 
100  times  as  great  as  those  of  the  sun.  That  is,  a  star  nearly 
as  brilliant  as  one  of  the  Pleiades  might  be  only  one  hundredth 
as  dense  as  the  sun  if  its  mass  were  only  10  times  greater. 
A  star  10  times  as  great  in  mass  and  one  tenth  as  dense  as 
the  sun  would  be  460  times  as  luminous. 

It  can  be  seen  from  this  incomplete  discussion  that  in 
order  that  a  star  shall  have  high  temperature  and  great 
luminosity  it  must  have  a  mass  at  least  as  great  as  that  of 
the  sun;  for  it  is  not  probable  that  a  much  denser  body 
would  be  in  a  gaseous  condition.  But  the  luminosity  of  a 
gaseous  star  is  so  sensitive  a  function  of  its  mass  that  one 
10  times  more  massive  than  the  sun  would  be  a  brilliant 
object  unless  its  density  were  exceedingly  low ;  and  one  only 
one  tenth  as  massive  as  the  sun  would  be  relatively  faint, 
even  if  it  were  as  dense  as  the  sun.  Therefore,  it  is  not 
strange  that  no  stars  with  very  small  masses  have  been 
found ;  one  as  small  as  one  of  the  planets  could  not  be  self- 


542    AN   INTRODUCTION   TO  ASTRONOMY  [ch.  xiii,  298 

luminous  while  in  a  gaseous  state.  On  the  other  hand,  no 
star  many  times  more  massive  than  the  sun  has  been  found. 
Perhaps  the  reason  is  that  the  data  respecting  masses  is  yet 
so  meager;  perhaps  the  temperatures  in  massive  stars  be- 
come so  great  that  their  atoms  disintegrate  and  the  remains 
fly  away  into  space. 

(d)  It  is  assumed  that  the  contraction  of  nebulas  into  stars 
began  at  such  a  time,  or  at  such  times,  and  that  the  individual 
nebulce  had  such  masses  that  there  has  resulted  the  present 
sidereal  system  of  nebulce  and  stars  in  all  stages  from  hottest  to 
''  coldest.  The  implications  of  this  assumption  are  not  at  once 
fully  evident;  they  can  be  brought  out  only  by  a  mathe- 
matical discussion  whose  results  alone  can  be  given  here. 

On  the  basis  of  Stefan's  law  of  radiation  and  the  assump- 
tion that  the  heat  of  a  star  is  developed  entirely  by  contrac- 
tion, it  is  found  that  the  change  of  radius  is  directly  propor- 
tional to  the  product  of  the  time  and  the  square  of  the  mass. 
If  there  are  other  important  sources  of  heat,  and  if  the 
radiation  is  from  a  layer  of  varying  depth  instead  of  from 
the  surface,  the  law  may  be  much  in  error.  But  on  the 
assumption  that  this  result  applies  .to  the  sun,  it  is  possible 
to  compute  the  time  required  for  it  to  have  contracted  from 
any  given  dimensions.  According  to  the  contraction  theory 
its  radius  is  now  diminishing  at  the  rate  of  a  mile  in  44  years. 
Consequently,  on  this  basis  it  has  contracted  from  the  orbit 
of  Mercury  in  1,500,000,000  years.  At  first  thought  this 
would  seem  to  give  a  long  supply  of  heat  to  the  earth  to 
meet  geological  needs;  but  if  the  sun  ever  filled  a  sphere 
as  large  as  the  orbit  of  Mercury  and  radiated  according  to 
Stefan's  law,  whatever  the  source  of  heat  may  have  been, 
its  temperature  must  have  been  so  low  that  its  rate  of  radia- 
tion could  have  been  only  a  little  more  than  one  seven- 
thousandth  that  at  present,  a  quantity  altogether  inade- 
quate to  support  Hfe  on  the  earth.  According  to  this  con- 
traction theory,  4,400,000  years  ago  the  radius  of  the  sun 
was  100,000  miles  greater  than  at  present,  and  its  rate  of 


CH.  XIII,  299]       THE   SIDEREAL  UNIVERSE  543 

radiation  was  only  two  thirds  that  which  is  now  observed. 
With  this  rate  of  radiation  the  theoretical  mean  temperature 
of  the  earth,  determined-  by  the  method  used  for  Mars  in 
Art.  172,  comes  out  51°  lower  than  at  present  (60°  F;),  or 
23°  below  freezing. 

The  second  part  of  the  law  gives  the  interesting  and  un- 
foreseen result  that  the  more  massive  a  star,  the  more  rapidly 
it  contracts.  Or,  if  the  results  are  translated  over  into  a 
relation  between  density  and  time,  it  is  found  that  if  a  star 
of  large  mass  and  one  of  smaller  mass  start  with  the  same 
density,  the  density  of  the  large  star  will  increase  faster 
than  that  of  the  smaller  one.  The  rate  of  change  of  density 
is  proportional  to  the  cube  root  of  the  fifth  power  of  the  mass. 
Therefore,  if  one  star  has  8  times  the  mass  of  another  and 
they  start  contracting  from  the  same  density,  it  will  arrive 
at  some  greater  density  in  -^  of  the  time  required  by  the 
smaller  star  to  reach  the  same  density.  As  applied  to  the 
stellar  system,  this  means  that  if  the  stars  all  started  con- 
densing from  nebulae  at  the  same  time,  those  which  have 
the  largest  masses  are  at  present  by  far  the  densest  and 
hottest.  The  large  stars  are  probably  much  hotter  on  the 
average  than  the  small  ones,  but  it  is  doubtful  if  they  are 
denser.  It  must  be  remembered  that  these  results  depend 
upon  the  very  questionable  assumption  that  the  heat  of 
stars  is  due  entirely  to  their  contraction. 

299.  The  Origin  and  Evolution  of  Binary  Stars.  —  The 
great  number  of  binary  stars  calls  for  a  consideration  of 
their  origin  and  evolution.  If  the  stars  have  condensed 
from  nebuke,  it  is  natural  to  suppose  that  binary  stars  have 
developed  from  nebulae  which  divided  into  two  parts,  or 
that  the  divisions  have  taken  place  after  the  condensing 
masses  have  reached  the  star  stage.  It  is  also  conceivable 
that  stars  which  originated  separately  have  later  united  to 
form  physical  systems.  Both  of  these  theories  will  be  con- 
sidered. 

Consider  first  the  theory  that  the  binary  stars  have  orig- 


544    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  299 

inated  by  the  fission  of  nebulae  or  larger  stars.  The  basis 
for  the  theory  is  the  very  reasonable  assumption  that  the 
original  nebula  had  more  or  less  rotation,  possibly  quite 
irregular  in  character.  In  those  cases  where  the  amount  of 
rotation,  that  is,  the  moment  of  momentum,  was  small,  it 
is  beheved  that  single  stars  rotating  slowly  have  resulted. 
In  those  cases  where  the  moment  of  momentum  was  large, 
it  is  supposed  that  there  has  been  separation  into  two  parts. 

There  is  some  theoretical  basis  for  this  conclusion,  though 
from  a  practical  point  of  view  it  has  generally  been  greatly 
overestimated.  In  a  brilhant  piece  of  work  on  figures  of 
equilibrium  of  homogeneous  fluids  rotating  as  solids,  Poin- 
care,  following  Maclaurin  and  Jacobi,  showed  that  for  slow 
rotation  an  oblate  spheroid  is  a  figure  of  equilibriima,  for 
faster  rotation  an  elongated  ellipsoid  is  the  corresponding  fig- 
ure, and  for  still  faster  rotations  the  ellipsoid  has  a  constric- 
tion, suggesting  that  for  still  faster  rotations  the  figure  would 
be  two  very  unequal  masses.  Now,  when  a  nebula  or  a  star 
contracts  it  rotates  more  rapidly  because  the  moment  of 
momentum  is  constant.  Hence  it  seems  reasonable  to  sup- 
pose that  nebulae  and  stars  follow  at  least  roughly  the  figures 
found  by  Poincare  for  the  homogeneous  case. 

There  is  one  very  important  point  of  difference  in  the  prob- 
lem treated  by  Poincare  and  that  presented  by  contracting 
bodies.  Poincare  considered  masses  all  of  the  same  density, 
but  having  different  rates  of  rotation.  In  a  contracting 
nebula  or  star  both  the  density  and  the  rate  of  rotation 
change.  The  increase  in  density  tends  to  sphericity;  the 
increase  in  rate  of  rotation  tends  to  oblateness.  The  two 
effects  almost  balance  each  other,  but  the  effect  of  increas- 
ing rotation  prevails  by  a  narrow  margin.  For  example, 
if  the  sun  contracts  with  loss  of  heat,  it  will  not  become  so 
oblate  as  Saturn  is  now  until  its  density  is  hundreds  of  times 
greater  than  that  of  platinum.  This  does  not  mean  that  a 
body  contracting  from  a  nebula  may  not  divide  into  two 
parts  at  any  stage  of  its  development,  but  it  shows  that  the 


CH,  xiii,  299]       THE    SIDEREAL   UNIVERSE  545 

tendency  for  fission  is  very  much  smaller  thaii  has  been 
supposed. 

Suppose  a  star  divides  into  two  parts.  Originally  the 
two  components  will  be  rotating  so  as  to  keep  their  same 
faces  toward  each  other.  But  with  further  contraction  they 
will  rotate  more  rapidly  while  their  period  of  tevolution  re- 
mains unchanged.  Then  tidal  evolution  begins,  and  under 
these  conditions  Darwin  has  shown  that  the  tides  will  in- 
crease the  periods  of  rotation  rapidly  and  the  period  of  revo- 
lution more  slowly.  Moreover,  if  the  original  orbit  had  any 
eccentricity  it  will  be  increased.  Consequently,  as  the  age 
of  a  binary  star  having  originated  by  fission  increases,  its 
period  of  revolution  increases  and  the  eccentricity  of  its 
orbit  increases. 

From  an  extensive  study  of  the  orbits  of  spectroscopic 
and  visual  binaries,  Campbell  has  found  that  stars  of  Types 
B  and  A  have  short  periods  and  nearly  circular  orbits,  and 
that  both  the  periods  and  the  eccentricities  increase,  on  the 
average,  through  the  spectral  types  F,  G,  K,  and  M.  One 
would  be  tempted  to  infer,  in  accordance  with  the  theory 
of  the  evolution  of  stars  through  the  spectral  types  from  B 
to  M,  that  binaries  of  Type  B  had  recently  originated  by^ 
fission  and  that  with  increasing  age  they  would  go  through 
the  various  spectral  types  with  periods  increasing  corre- 
spondingly from  a  few  hours  to  an  average  of  more  than  a 
century,  and  the  eccentricity  from  near  zero  to  an  average 
of  about  0.5. 

But  such  an  inference  would  be  entirely  unwarranted  and 
erroneous,  for  an  ample  consideration  of  the  dynamics  in- 
volved shows  that  when  a  nebula  or  star  divides  into  two 
equal  masses,  tidal  friction  in  any  time  however  long  is  not 
competent  to  make  the  period  more  than  ^bout  twice  its 
original  value;  if  the  masses  are  unequal  but  comparable, 
as  in  the  case  of  all  known  binaries,  the  period  may  be 
lengthened  several  fold.  But  it  is  altogether  impossible  for 
tidal  friction  to  increase  the  period  of  a  binary  star  whose 
2n 


546    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  299 

components  have  comparable  masses  from  a  few  hours  or 
days  to  the  many  years  found  in  the  case  of  most  visual 
binaries. 

There  is  a  similar  difficulty  in  the  eccentricities  of  the 
orbits  of  binary  stars.  Consequently  the  important  facts 
brought  out  in  Campbell's  discussion  do  not  confirm  the 
current  theory  of  the  -evolution  of  the  stars.  So  far  as  the 
periods  are  concerned  they  are  in  harmony  with  the  hy- 
pothesis that  the  B  and  A  stars  are  massive,  for  the  greater 
the  mass,  the  shorter  the  period  for  a  given  distance  between 
the  stars,  but  it  is  highly  improbable  that  the  great  range  of 
periods  depends  upon  the  masses  alone.  The  dynamical 
conditions  imply  that  if  visual  binaries  originated  by  fission, 
the  division  took  place  while  they  were  yet  in  the  nebular 
stage. 

The  hypothesis  that  two  independent  stars  can  unite  to 
form  a  binary  remains  to  be  considered.  If  two  stars  are 
drawn  toward  each  other  by  their  mutual  gravitation,  they 
may  pass  near  and  around  each  other  without  any  contact, 
as  a  comet  passes  around  the  sun ;  each  may  collide  with 
the  outlying  parts  of  the  other ;  they  may  undergo  a  grazing, 
or  partial,  collision;  and,  in  the  extreme  case,  they  may 
have  a  central  colhsion.  If  they  do  not  collide  at  all,  they 
will  recede  to  the  distance  from  which  they  were  drawn 
together,  and  a  binary  star  cannot  result.  If  they  suffer  a 
collision  with  outlying  parts,  their  velocities  will  be  reduced 
and  they  may  not  recede  to  a  very  great  distance  from  each 
other.  The  character  of  their  orbits  after  collision  will  de- 
pend upon  the  amount  of  kinetic  energy  which  is  trans- 
formed at  the  time  of  collision.  This  energy  goes  into  heat, 
and  the  question  arises  whether,  if  sufficient  motion  is  de- 
stroyed to  produce  a  binary,  the  heat  evolved  may  not  reduce 
both  stars  to  the  nebulous  state. 

Consider  a  special  example  of  two  stars  each  in  mass 
equal  to  the  sun.  At  a  great  distance  from  each  other  their 
relative  velocities  might  be  anything  from  zero  to  several 


CH.  xm,  299]       THE    SIDEREAL   UNIVERSE 


547 


hundred  miles  per  second ;  take  the  most  favorable  case  where 
it  is  zero.  Suppose  that  at  their  nearest  approach  their 
distance  from  each  other  is  as  great  as  that  from  the  earth 
to  the  sun.  Under  the  hypotheses  adopted  they  will  have 
a  relative  velocity  of  about  37  miles  per  second.  Suppose 
they  encounter  enough  resistance  from  outlying  nebulous 
or  planetesimal  matter,  or  from  coUision  with  a  planet,  to 
reduce  their  most  re- 
mote point  of  recession 
after  collision  to  100 
astronomical  xmits. 
It  can  be  shown  that 
their  velocity  must 
have  been  reduced  by 
sij!  of  its  amount,  or 
by  0.185  mile  per 
second.  This  would 
generate  as  much  heat 
as  the  sun  radiates  in 
about  8  years.  Conse- 
quently the  expansive 
effect  of  the  heat 
generated  by  the  col- 
hsion  will  not  be  im- 
portant, and  after  the 
encounter  the  stars 
will  be  moving  in  an  orbit  whose  eccentricity  is  0.98  and 
whose  period  is  about  250  years.  The  resistance  could  have 
been  produced  by  collision  with  a  planet  whose  mass  was  ^^ 
that  of  one  of  the  suns.  It  follows  that  if  a  star  passing  the 
sun  should  meet  Jupiter,  something  comparable  to  what  has 
been  given  in  the  example  wou'.d  result.  Figure  186  shows 
the  original  parabola,  the  point  of  collision  P,  and  the 
elliptical  orbit  after  collision. 

Now  let  us  follow  out  the  history  of  the  star  after  such  a 
collision  as  has  been  described.     If  there  are  no  subsequent 


Fig.  186.  —  Reduction  of  parabolic  orbit  to 
an  ellipse  by  collision  of  a  sun  with  a  planet 
of  another  sun. 


548    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  299 

collisions,  the  stars  will  continue  to  describe  very  elongated 
elliptical  orbits  about  their  center  of  gravity.  If  there  are 
subsequent  collisions  with  other  planets  or  with  any  other 
material  in  the  vicinity  of  the  stars,  their  points  of  naarest 
approach  will  not  be  appreciably  changed  unless  the  colU- 
sions  are  far  from  the  perihelion  point,  their  points  of  most 
remote  recession  will  be  diminished  by  each  collision,  and 
the  result  is  that  both  the  period  and  the  eccentricity  of  the 
orbit  will  be  decreased  as  long  as  the  process  continues.  If 
this  is  the  correct  theory  of  the  origin  of  binary  stars,  those 
whose  periods  and  eccentricities  are  small,  are  older  on  the 
average,  at  least  as  binaries,  than  those  whose  periods  and 
eccentricities  are  large,  and  this  would  suggest  that  the  B 
and  A  stars  are  older  than  the  K  and  M  stars.  The  only 
obvious  difficulty  with  the  basis  of  this  theory  of  the  origin 
of  binary  stars  is  that  these  near  approaches  and  partial 
collisions  are  necessarily  extremely  infrequent,  while  binary 
stars  are  very  numerous.  The  seriousness  of  this  difficulty 
depends  upon  the  length  of  time  the  stars  endure,  about 
which  nothing  certain  is  known. 

As  has  been  stated  in  Art.  294,  a  central  colhsion  would 
produce  a  temporary  star,  which  would  later  change  into  a 
nebula. 

300.  The  Question  of  the  Infinity  of  the  Physical  Uni- 
verse in  Space  and  in  Time  — There  are  transcendental 
questions  which,  from  their  nature,  can  never  be  answered 
with  certainty,  but  which  the  human  mind  ever  persists  in 
attacking.  Among  such  questions  is  that  of  the  infinity  of 
the  physical  universe  in  space  and  in  time. 

It  has  been  seen  in  Art.  270  that  the  apparent  distribu- 
tion of  the  stars  proves  that  they  cannot  be  scattered  uni- 
formly throughout  infinite  space.  It  has  also  been  seen 
that  there  is  no  observational  evidence  that  galaxies,  sep- 
arated by  distances  of  a  higher  order  than  those  between  the 
stars,  may  not  be  units  in  larger  aggregations  and  so  on  to 
super-galaxies  without  limit.     This  may  be  adopted  as  a 


CH.  XIII,  300],      THE   SIDEREAL  UNIVERSE  549 

working  hypothesis.  We  may  then  inquire  whether  there 
will  be  luminous  stars  through  infinite  time,  or  whether  they 
all  will  ultimately  become  extinct. 

According  to  physical  laws  as  they  are  known  at  present, 
the  stars  are  pouring  radiant  energy  out  into  the  ether  at 
an  extravagant  rate  and  it  is  not  being  returned  to  them  in 
relatively  appreciable  amounts.  For  example,  the  sun  loses 
more  hght  and  heat  by  radiation  in  a  second  than  it  will 
receive  from  aU  the  stars  in  the  sky  in  a  million  years.  It  is 
inconceivable  that  a  star  has  an  unlimited  store  of  internal 
energy.  Therefore  its  energy  will  ultimately,  become  ex- 
hausted unless  a  new  supply  is  furnished  in  some  way.  One 
method  by  which  the  internal  energy  of  a  star  may  be  in- 
creased is  by  collision  with  another  star.  But  after  colli- 
sion the  combined  mass  would  lose  its  energy  similarly  until 
another  restoration  by  another  collision.  But  by  this  pro- 
cess the  matter  of  the  universe  becomes  aggregated  in 
larger  and  larger  masses,  and  if  it  is  finite  in  amount,  a 
stage  will  be  reached  when  no  more  collisions  will  take  place. 
Then  these  final  Stars  will  in  the  course  of  time  radiate  away 
all  their  internal  energy  and  remain  throughout  eternity 
dark,  cold,  and  hfeless.  At  least,  such  is  the  teaching  of 
present-day  science  if  the  physical  universe  is  finite,  as  has 
usually  been  assumed. 

But  now  suppose  that  there  are  myriads  of  galaxies  compos- 
ing larger  and  still  larger  cosmic  units,  and  remember  that 
there  are  no  observational  facts  whatever  which  contradict 
this  hypothesis.  Under  this  assumption  the  energy  in  the 
universe  is  also  infinite.  It  does  not  follow  from  this, 
however,  that  it  will  last  an  infinite  time,  for  there  are,  by 
hypothesis,  infinitely  many  bodies  which  are  subject  to 
collisions  and  which  are  radiating  energy  into  the  ether. 
But,  on  the  other  hand,  if  the  relative  speed  of  the  larger 
cosmic  units  is  great  enough,  there  will  be  enough  energy  to 
last  the  infinite  universe  an  infinite  time.  This  follows  from 
the  fact  that  infinities  may  be  of  different  orders,  as  the 


550    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  300 

mathematicians  say.  The  actual  demands  in  the  present 
case  are  not  severe.  In  order  that  the  energy  should  last 
an  infinite  time  it  is  sufficient  that  the  relative  speeds  of  the 
larger  cosmic  units  of  all  order  shall  exceed  some" finite  value. 
The  energy  in  any  particular  galaxy  might  run  down,  as 
in  the  finite  case  considered  above ;  but,  according  to  the 
present  hypothesis,  at  immense  intervals  this  galaxy  wOuld 
collide  with  some  other  one  with  speed .  sufficient  to  restore 
its  internal  energies  if  the  energy  of  their  relative  motions 
were  thus  transformed.  It  might  require  only  a  veiry-small 
fraction  of  the  energy  of  the  relative  motions.  The  prrocess 
would  terminate,  however,  if  there  were  only  a  finite  number 
of  galaxies,  but  by  hypothesis  the  super-galaxies  are  units 
in  still  larger  aggregations.  There  might  be  a  restoration 
of  heat  energy  by  interactions  of  these  larger  units,  and  so 
on  without  Umit.  It  is  not  profitable  to  pursue  the  inquiry 
further  here,  but  it  is  not  withofit  interest  to  know  that 
according  to  our  present  understanding  of  the  laws-of  nature 
it  is  not  necessary  to  conclude  that  the  physical  universe 
will  in  a  finite  time  reach  the  condition  of  eternal  ni^t  and 
death.  This  discussion  also  gives  an  answer,  though  perhaps 
not  the  correct  one,  to  the  question  why  the  universe  h^  not 
already  attained  a  condition  of  stagnation  and  death.  In 
short,  it  gives  a  picture  of  a  universe  whose  Hfe  and  activity 
are  without  beginning  and  without  end. 


IV.   The  Nebula 

301.  Irregular  Nebulae.  —  There  are  many  nebulae  in 
the  sky  of  enormous  extent  and  irregular  form.  Among  the 
finest  examples  of  these  objects,  though  by  no  means  the 
most  extensive,  are  the  veil-like  structures  which  are  seen  in 
the  constellation  Cygnus,  one  of  which  is  shown  in  Fig.  187. 
It  is  altogether  probable  that  they  are  at  least  as  remote  as 
the  nearer  stars.  Since  they  extend  across  regions  occupied 
by  hundreds  of  stars,  they  are  of  inconceivable  magnitude ; 


CH.  XIII,  301]       THE    SIDEREAL  UNIVERSE 


551 


Fig.  187.  —  Irregular  nebula  in  Cygnus  (N.  G.  C.  6960).     Photographed  by 
Ritchey  with  the  two-foot  reflector  of  the  Yerkes  Observatory. 


552    AN   rNTRODUCTION   TO  ASTRONOMY  [ch.  xiii,  301 

certainly  a  hundred  years  are  required  for  light  to  cross  them. 
They  are  extremely  faint  (the  long-exposure  photographs 
being  quite  misleading)  and  they  are  probably  very  tenuous, 
though  nothing  is  actually  known  regarding  their  density. 
If  they  are  condensing  under  gravitation,  the  process  must 
be  going  on  extremely  slowly. 

An  example  of  a  less  widely  extended  and  apparently 
much  denser  nebula  is  the  great  nebula  in  Orion  (Fig.  61), 
which  is,  perhaps,  the  most  wonderful  and  beautiful  object 
in  the  heavens.  It  fills  a  space  whose  apparent  diameter 
is  more  than  half  a  degree.  This  means  it  is  of  enormous 
volume,  for  it  is  as  remote  as  certain  stars  which  are  asso- 
ciated with  its  denser  parts.  Its  parallax  can  scarcely  be 
over  0".01  and  it  probably  is  much  smaller;  if  the  larger 
value  is  correct,  its  diameter  is  20,000,000  times  that  of  the 
sun  and  several  years  would  be  required  for  light  to  travel 
from  one  side  of  it  to  the  other.  The  density  of  the  Orion 
nebula  is  altogether  unknown,  but  it.  is  generally  regarded 
as  being  very  low.  If  it  averages  even  xjnr/insir  that  of  the 
atmosphere  and  if  it  is  spherical  ( ?),  its  total  mass  is 
100,000,000,000,000  times  that  of  the  sun,  and  in  spite  of  its 
enormous  distance,  its  attraction  for  the  earth  is  one  fourth 
that  of  the  sim.  If  the  nebula  is  rare,  it  is  difficult  to  account 
for  its  radiation,  because  it  could  not  have  a  high  temperature 
except  possibly  in  its  deep  interior  where  pressure  of  the  out- 
lying parts  would  prevent  expansion.  The  luminosity  of  the 
nebulae,  like  that  of  the  comets,  has  long  been  an  xmexplained 
phenomenon. 

The  form  of  the  Orion  nebula  suggests  whirling  motions 
of  its  parts.  Relative  internal  motions  were  foimd  first 
by  Bourget,  Fabry,  and  Buisson;  Frost  and  Maney  have 
shown  by  the  spectroscope  that  its  northeastern  part  is 
receding  from  the  solar  system,  while  the  Southwestern  part 
is  approaching  at  the  relative  rate  of  about  6  miles  per  second. 
It  is-  clear  that  imless  the  density  is  sufficiently  great  these 
motions  will  cause  the  nebula  to  dissipate  in  space.     On  the 


CH.  xni,  301]       THE    SIDEREAL  UNIVERSE  653 

assumption  that  this  is  simply  a  motion  of  rotation,  and 
neglecting  gaseous  expansion,  it  is  found  that  the  nebula  is 
in  no  danger  of  disrupting  if  its  average  density  is  greater 
than  10"^^  times  that  of  water.  At  this  hmiting  density  its 
total  mass  would  about  equal  that  of  the  sun. 

It  was  supposed  in  the  days  of  Sir  William  Herschel  that 
the  nebulae  may  be  galaxies  which  are  so  remote  that  their 
individual  stars  are  not  distinguishable,  even  with  the 
most  powerful  telescopes.  This  is  certainly  not  the  true 
explanation  of  the  irregular  nebulae.  In  the  first  place,  the 
spectra  of  the  brighter  ones  for  which  the  data  are  at  hand 
consist  of  bright  Unes,  proving  on  the  basis  of  the  first  law 
of  spectrum  analysis  that  they  are  incandescent  gases  under 
low  pressure.  The  bright  Unes  belong  to  a  hypothetical  ele- 
ment nebuhum,  found  only  in  nebulse,  and  to  hydrogen.  In 
the  second  place,  they  are  condensed  in  the  zone  of  the  Milky 
Way,  which  indicates  they  are  in  some  way  connected  with 
it.  Campbell  and  Moore  have  found  that  they  show  the 
streaming  tendencies  which  are  characteristic  of  the  stars. 
For  these  reasons  the  conclusion  is  held  that  they  are  tenuous 
gaseous  members  of  our  own  Galaxy. 

A  very  interesting  fact  has  recently  been  discovered  in 
connection  with  the  Magellanic  Clouds,  two  masses  of 
stars  in  the  far  southern  heavens,  having  the  appearance  of 
two  smaller  galaxies  which  are  quite  independent  of  the 
Milky  Way.  R.  E.  Wilson,  at  the  South  American  branch 
of  the  Lick  Observatory,  has  found  that  the  radial  velocities 
of  the  nebulae  in  the  Magellanic  clouds  which  are  bright 
enough  for  measurement  show  rapid  recession  of  all  of  these 
objects,  the  average  speed  being  over  150  miles  per  second. 
This  suggests  that  these  aggregations  of  stars  have  velocities 
with  respect  to  our  own  Galaxy  of  a  higher  order  than  the 
average  internal  velocities,  in  harmony  with  the  suggestion 
in  Art.  300. 

Barnard  has  recently  brought  forward  strong  evidence 
for  the  conclusion  that  there  are  relatively  dark  and  opaque 


554    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  301 

masses,  perhaps  nebulous  in  character,  in  certain  parts  of  the 
Milky  Way.  He  has  found  regions  in  which  the  stars  seem 
to  be  blotted  out  by  obscure  material,  as  is  shown  in  Fig. 
188.  Probably  the  apparent  breaks  in  some  of  the  nebulae, 
as,  for  example,  the  celebrated  Trifid  Nebula  in  Sagittarius 
(Fig.  189),  are  due  to  obscuring  material  which  cuts  off  the 
hght  from  certain  regions.     At  any  rate,  it  is  difficult  to  see 


Fio.  188.  —  Oij  the  left  a  bright  nebula  (in  Cygnus)  and  on  the  right  a 
dark  patch  which  is  probably  due  to  a  dark  nebula.  Photographed  by 
Barnard  at  the  Yerkes  Observatory. 

how  matter  could  be  in  equiUbrium  in  any  such  forms  as  the 
luminous  matter  assumes. 

302.  Spiral  Nebulae.  —  Spiral  nebulae  are  more  numerous 
than  all  other  kinds  together.  According  to  Keeler's 
original  estimate  there  are  at  least  120,000  within  the  reach 
of  the  telescope  which  he  used;  there  may  be  five  or  ten 
times  the  number  within  reach  of  the  great  reflectors  of  the 
Solar  Observatory  of  the  Carnegie  Institution.  They  are 
characterized  by  their  great  extent  (Fig.  190)  and  by  irregular 
arms,  generally  two  in  number  when  they  are  distinctly  de- 
fined, which  wind  out  from  centers.  They  almost  invariably 
have  well-defined  centers,  apparently  of  considerable  den- 
sity, and  their  arms  usually  contain  a  number  of  conspicuous 
local  condensations,  or  nuclei. 


CH.  xm,  302]       THE    SIDEREAL  UNIVERSE 


555 


The  spiral  nebulse  are  further  characterized  by  being  white, 
whereas  the  large  irregular  nebulae  have  a  greenish  tinge  due 
to  the  green  light  from  nebulium.  Most  of  them  are  too 
faint  for  detailed  spectroscopic  study,  but  some  of  the 
brighter  of  them  have  been  found  to  have  spectra  similar 
to  the  sun's  spectrum.  This  leads  to  the  inference  that  they 
are  perhaps  partly  solid  or  liquid-  On  the  other  hand, 
Seares  has  photo- 
graphed some  of 
them  through  a 
screen  which  cuts 
off  the  blue  end 
of  the  spectrum. 
The  brightness  of 
the  arms  was 
much  more  re- 
duced than  that 
of  the  central 
nuclei,  indicating 
that  a  consider- 
able part  of  their 
light  is  similar  to 
that  from  gases. 
Moreover,  their 
transparency  im- 
plies that  they  are 
tenuous.    Hence, 

they  seem  to  be  vast  swarms  of  incandescent  sohd  or  Uquid 
particles,  perhaps  with  many  larger  masses,  surrounded  by 
gaseous  materials.  There  is  difficulty  in  explaining  their 
luminosity,  though  Lockyer  attempted  to  account  for  the 
light  of  all  nebulae  by  ascribing  it  to  heat  generated  by  the 
collisions  of  meteorites  of  which  he  supposed  they  are  largely 
•  composed.  The  obscure  material  in  and  around  nebulae 
may  be  very  abundant.  This  supposition  is  confirmed  in  the 
case  of  spiral  nebulae,  for  when  one  is  seen  edgewise  the  dark 


Fig.  189.  — The  Trifid  Nebula.  The  dark  lanes 
by  which  it  is  crossed  are  probably  due  to  inter- 
vening dark  material.      Photographed  with  the 

■    Crossley  reflector  of  the  Lick  Observatory. 


556    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  302 

material  at  its  periphery  eclipses  the  center  and  causes  an 
apparently  dark  rift  through  it  (Fig.  191).  Another  dis- 
tinguishing feature  of  spiral  nebulse  is  that  they  are  very 
infrequent  in  or  near  the  Milky  Way. 


Fig.  190.  —  Spiral  nebula  in  Ursa  Major  fM.  101).    Photographed  by  Ritchey 
at  the  Yerkes  Observatory. 

The  spiral  nebulae  range  in  magnitude  all  the  way  from  the 
Great  Nebula  in  Andromeda  (Fig.  192),  which  is  about  1°.5 
long  and  30'  wide,  to  minute,  faint  objects, which  are  barely 
discoverable  after  long  exposures  with  powerful  photographic 
telescopes.  There  is  no  reason  to  beUeve  there  are  not  others 
still  smaller.     Since  the  Andromeda  nebula  is  certainly  as 


CH.  xiii,  302]       THE   SIDEREAL  UNIVERSE 


557 


distant  as  the  nearest  stars,  its  volume  is  enormous ;  the 
smallest  ones  may  be  as  small  as  the  solar  system,  though  they 
would  wind  up  and  lose  their  spiral  characteristics  in  a  short 
time. 

The  suggestion  has  been  made  (Art.  249)  that  a  spiral 
nebula  may  develop  when  a  star  is  visited  closely  by  another 
star,  or  when  a  group  of  stars  passes  near  another  group  of 
stars.  There  is  no  apparent  difficulty  in  explaining  small 
spirals  in  this  way,  but  the 
large  ones  present  a  more 
serious  problem,  especially 
if  we  Umit  ourselves  to  the 
close  approach  of  two  single 
stars.  It  is  not  at  all  neces- 
sary to  do  this,  for  in  a 
general  way  the  dynamical 
principles  involved  apply  to 
aggregates  of  all  dimensions 
up  to  galaxies,  and  even 
beyond  if  there  are  larger 
units  in  the  tmiverse.  There 
is  possibly  some  evidence 
that  the  Milky  Way  has  a 
spiral  structure. 

Although  the  larger  spirals 
are  enormous  in  extent,  they 
may  have  only  moderate  masses.  However  improbable 
this  may  be  on  the  basis  of  their  appearance,  it  must  be  re- 
membered that  there  is  no  direct  evidence  whatever  at 
present  regarding  their  masses,  and  the  source  of  their  lumi- 
nosity is  quite  unknown.  It  is  natural  to  suppose  that 
though  a  spiral  of  dimensions  comparable  to  the  solar  system 
might  be  produced  by  the  disruptive  forces  of  a  near  approach 
of  two  stars,  it  would  not  be  possible  for  one  a  thousand 
times  larger  to  be  formed  in  the  same  way.  An  exami- 
nation of  the  equations  involved  shows  that,  if  a  certain 


Fig.  191.  —  Spiral  nebula  in  Androm- 
eda (H.  V.  19)  presenting  edge 
toward  the  earth.  Central  line 
eclipsed  by  obscure  material.  Pho- 
tographed with  the  Crossley  reflector 
of  the  Lick  Observatory. 


558    AN   INTRODUCTION   TO   ASTRONOMY   [ch.  xiii,  302 

velocity  of  ejection  would  cause  matter  to  recede  (neglect- 
ing the  attraction  of  the  passing  sun)  to  the  distance  of 
Neptune,  a  velocity  one  twenty-four-thousandth  greater 
would  cause  it  to  recede  1000  times  farther  (Table  XIII). 
Hence  the  argument  against  very  large  spirals  being  formed 
by  the  near  approach  of  two  great  sims  is  not  so  conclusive 
as  it  might  at  first  seem.  They  may  have  been  formed, 
however,  by  the  passage  near  one  another  of  two  great 
groups  of  stars  such  as  the  globular  clusters ;  or  they  may 
have  been  formed  in  some  other  way  not  yet  considered. 

The  spectra  of  spiral  nebulae  are  in  harmony  with  the 
suggested  mode  of  their  origin.  Their  distribution  demands 
consideration.  Their  apparent  distribution  may  mean  that 
they  are  out  on  the  borders  of  the  Galaxy  and  that  they, 
are  not  seen  in  the  Milky  Way  because  of  their  great  distances 
in  these  directions.  It  would  be  expected  that  close  ap- 
proaches would  occur  most  frequently  in  the  interior  of  the 
Galaxy  where  the  stars  move  the  fastest  if  they  are  making 
excursions  to  and  fro  through  it.  On  the  other  hand,  out  on 
the  borders  they  would  move  more  slowly  and  their  mutual 
attractions  would  be  more  efficient  in  bringing  them  to- 
gether. 

There  is  one  fact  which  is  opposed  to  the  suggested  ex- 
planation of  spiral  nebulae,  and  that  is,  as  Shpher  first  foimd, 
their  radial  velocities  average  very  great.  For  example,  the 
Great  Andromeda  Nebula  is  approaching  the  solar  system  at 
the  rate  of  200  miles  per  second.  Moreover,  Slipher  found 
spectroscopic  evidence  that  it  is  rotating.  Even  if  the  result 
is  in  doubt  for  this  nebula,  it  is  altogether  certain  in  the  case 
of  another  spiral  which  is  edgewise  to  the  earth,  and  which 
Slipher  investigated  in  1913.  Among  the  stars  high  veloc- 
ities are  on  the  whole  associated  with  small  masses.  If  this 
is  a  universal  principle,  which  seems  dynamically  sound, 
the  spirals  must  have  smaller  masses  than  any  known 
class  of  stars.  Or,  perhaps,  spirals  have  been  formed  on  the 
whole  only  from  stars  which  passed  one  another  at  great 


CH.  xm,  302]       THE    SIDEREAL  UNIVERSE 


559 


Fig.  192.  —  Great  Nebula  in  Andromeda.    Photographed  by  Ritchey  with  the 
two-foot  reflector  of  the  Yerkes  Observatory. 


560    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  302 


speed„and  they  of  course  still  possess  most  of  their  kinetic 
energy. 

It  has  been  more  than  once  suggested  that  the  spiral  neb- 
ulae are  not  in  reality  nebulae  at  all,  but  distant  galaxies. 
If  this  is  true,  it  is  difficult  to  explain  their  distribution  with 
respect  to  the  Milky  Way,  or  their  strong  central  condensa- 
tions, or  the  fact  that  they  are  crossed  by  dark  streaks  when 
they  are  presented  edgewise  to  us.  Besides,  the  results  of 
Seares'  photographs  are  opposed  to  this  hypothesis. 

303.  Ring  Nebulae.  —  A  few  nebulse  have  the  form  of 
almost  perfect  rings,  the  best  example  of  which  is  the  one 

between  Beta  Lyrse  and  Gamma 
Lyrse  (Fig.  193).  This  nebula  has 
a  fifteenth-magnitude  star  near  its 
center  which  has  been  suspected 
of  being  variable.  It  is  probably 
associated  with  the  nebula,  though 
this  is  not  certain.  The  spectrum 
of  the  ring  nebula  in  Lyra  has 
been  examined  and  it  has  been 
found  that  hydrogen  extends  out 
considerably  beyond  the  heUum. 
The  origin  and  development  of 
these  remarkable  objects  are  quite 
beyond  conjecture  at  present. 

304.  Planetary  Nebulae. —The 
planetary  nebulae  are  supposed  to 
be  next  to  the  0-type  stars  in  evolution,  and  the  0-type  stars 
are  supposed  to  precede  the  B-type  stars.  They  are  in  all 
cases  apparently  small  in  size,  usually  rather  dense,  particu- 
larly near  their  centers,  and  they  have  rather  well-defined 
outhnes.  They  were  named  by  Herschel  from  their  resem- 
blance to  faint  planetary  disks;- 

The  spectra  of  about  75  planetary  nebulae  have  been  ex- 
amined. Perhaps  the  most  important  result  of  this  examina- 
tion is  that  their  radial  velocities  (24  miles  per  second)  are 


Fig.  193.  —  The  ring  nebula 
in  Lyra.  Photographed  by 
Sullivan  at  the  Yerkes  Ob- 
servatory with  the  40-inch 
telescope. 


CH.  xm,  304]       THE    SIDEREAL  UNIVERSE 


561 


at  least  three  times  those  of  the  stars  of  Type  B.  This  is 
squarely  opposed  to  the  theory  that  they  condense  into  stars 
of  Types  O  and  B.  If  this  theqry  is  maintained,  an  explana- 
tion of  the  greatly  decreased  velocities  is  demanded,  and 
none  is  at  hand.  On  the  other  hand,  the  novse  go  first  into 
planetary  nebulae  and  then  into  Wolf-Rayet  stars. 

The  central  parts  of  planetary  nebulae  give  the  lines  of 
nebulivun  and  hydrogen;  the  outermost  parts  give  the 
hydrogen  lines  alone.  That  is,  hydrogen  forms  an  atmos- 
phere around  the  denser  nebulium 
and  hydrogen  cores. 

The  problem  of  the  rotation  of 
planetary  nebulae  is  now  being 
taken  up  at  a  number  of  observa- 
tories. By  an  adaptation  of  the 
spectroscope  first  employed  by 
Keeler  on  the  rings  of  Saturn,  and 
used  more  recently  by  Slipher  at 
the  Lowell  Observatory  on  planets 
and  spiral  nebulae,  Campbell  and 
Moore  have  found  that  two  of 
these  remarkable  objects  are  rotat- 
ing around  axes  approximately  at 
right  angles  to  a  plane  passing 
through  the  earth  and  the  longer  axes  of  the  nebulae.  On 
the  basis  of  the  observed  relative  velocities  of  3.1  to  3.7  miles 
per  second,  and  plausible  assumptions  regarding  the  distance 
of  the  nebulae,  they  found  that  their  masses  are  between  3  and 
100  times  that  of  the  sun,  with  periods  of  rotation  between 
600  and  14,000  years.  With  such  slow  rates  of  rotation  there 
is  no  possibility  of  these  objects  ever  dividing  into  two  parts 
and  forming  a  binary  star,  in  spite  of  the  ^fact  that  their 
density  probably  does  not  exceed  one  millionth  that  of  our 
atmosphere  at  sea  level. 


Fig.  194.  —  A  planetary  neb- 
ula. Photographed  with  the 
Crossley  reflector  at  the  Lick 
Observatory. 


2o 


562    AN   INTRODUCTION   TO  ASTRONOMY   [ch.  xiii,  304 


XXIV.    QUESTIONS 

1.  If  500,000,000  stars  were  scattered  uniformly  over  the  celes- 
tial sphere,  what  would  be  the  apparent  angular  distance  between 
adjacent  stars?  If  another  star  were  placed  at  random  on  the 
sky,  what  would  be  the  probability  that  it  would  be  within  1" 
of  one  of  these  stars? 

2.  In  the  part  of  the  sky  covered  by  Aitken's  survey  of  double 
stars  (north  of  decUnation  —14°)  there  are  about  200,000  stars 
brighter  than  the  tenth  magnitude ;  what  is  the  average  distance 
between  adjacent  members  of  this  list  \of  stars  ?  Aitken  found 
5400  pairs  separated  by  less  than '  5" ;  what  is  the  probability 
that  a  particular  one  of  these  oases  is  accidental?  What  is  the 
probability  that  they  are  all  accidental?  According  to  the  laws 
of  probability,  how  many  of  the  5400  stars,  in  a  random  arrange- 
ment, should  be  separated  less  than  5"  ? 

3.  Suppose  the  apparent  distance  between  two  stars  miist  be 
at  least  0".2  in  order  that  they  may  be  seen  as  two  distinct  stars 
with  the  largest  telescopes ;  suppose  the  distance  of  a  double  star 
is  500  parsecs ;  what  must  be  the  distance,  in  astronomical  units, 
between  the  components  in  order  that  they  may  be  seen  as  sepa- 
rate stars?  If  the  mass  of  each  star  is  equal  to  that  of  the  sun, 
what  will  be  their  period  of  revolution  (Art.  154)  ?  If  their  dimen- 
sions and  surface  brilliancy  are  the  same  as  those  of  the  sun,  what 
will  be  their  magnitude  taken  together  ? 

4.  Suppose  the  relative  velocity  of  the  two  components  of  a 
double  star  must  be  5  miles  per  second  in  order  that  it  may  be 
possible  to  determine  by  the  spectroscope  that  the  star  is  a  binary ; 
how  near  must  the  components  be  to  each  other  in  order  that  it 
may  be  possible  to  find  that  the  star  is  a  binary  if  their  combined 
mass  is  one  tenth  that  of  the  sun?  Equal  to  that  of  the  sun? 
Ten  times  that  of  the  sun? 

5.  Suppose  the  density  of  the  components  of  a  binary  star  is 
equal  to  that  of  the  sun  and  that  the  two  components  (assumed 
spherical)  are  in  contact ;  what  is  their  period  of  revolution  if 
their  combined  mass  is  one  tenth  that  of  the  sun?  Equal  to  that 
of  the  sun?  Ten  times  that  of  the  sun?  What  are  their  relative 
velocities  in  the  respective  oases?  What  are  their  temperatures 
in  the  respective  oases  [Art.  298  (c)]?  What  are  their  luminosities 
in  the  respective  oases  ? 

6.  Suppose  the  two  components  of  an  eclipsing  variable  are 
equal  in  mass  and  that  their  density  is  that  of  the  sun ;  what  is 
the  ratio  of  the  time  of  eclipse  to  the  period  of  revolution  if  their 


CH.  XIII,  304]       THE    SIDEREAL   UNIVERSE  563 

combined  mass  is  one  tenth  that  of  the  sun?  Equal  to  that  of 
the  sun?  Ten  times  that  of  the  sun?  Solve  the  problem  if  their 
density  is  one  tenth  that  of  the  sun,  and  also  if  it  is  ten  times  that 
of  the  sun. 

7.  Which  of  the  ten  phenomena  of  Art.  296  fail  to  arrange  the 
stars  strictly  in  the  order  B,  A,  P,  G,  K,  M?  Which  of  the  ten 
phenomena  are  opposed  to  the  hypothesis  that  the  spectral  type 
of  a  star  depends  on  its  mass?  Which  of  the  ten  phenomena  are 
opposed  to  the  hypothesis  that  the  arrangement  of  stars  according 
to  age  is  M,  A,  B,  A,  F,  G,  K,  M  (the  hypothesis  of  Lockyer  and 
Russell)  ? 

8.  The  apparent  areas  of  the  sun  and  the  denser  part  of  the 
Orion  nebula  are  about  the  same,  and  the  sun  is  about  30  magnitudes 
brighter  than  the  nebula.  Suppose  the  amount  of  light  they  radiate 
is  proportional  to  the  fourth  powers  of  their  absolute  tempera- 
tures. What  is  the  temperature  of  the  Orion  nebula?  If  its 
diameter  is  20,000,000  times  that  of  the  sun,  what  is  its  mass 
(computed  from  the  relation  connecting  temperature,  mass,  and 
density  of  a  gaseous  body)?  Under  the  same  assumptions,  what 
is  its  mean  density?  (The  student  will  not  fail  to  remember  that 
some  of  the  assumptions  on  which  the  computation  rests  are  ques- 
tionable.) 


INDEX  OF  NAMES 


Abbott,  268,  350,  351,  380 
Adams,  J.  C,  240,  241,  257 
Adams,  W.  S.,  388,  389 
Agenor,  160 
Airy,  240 
Aitken,  506,  507 
Albategnius,  117 
Aldrich,  350,  351 
Alexander  the  Great,  116 
Anderson,  523 
Angstrom,  371,  390 
Antoniadi,  285,  286 
Areas,  151 
Argelander,  139 
Aristarchus,  41,  79,  116 
Aristotle,  40,  79,  116 
Arrhenius,  73,  403 

Backhouse,  263 

Bacon,  Boger,  6 

Bailey,  522 

Baily,  62 

Barnard,  260,  263,  278,  285,  289, 
291,  293,  299,  300,  301,  302, 
305,  308,  312,  325,  327,  331, 
335,  337,  402,  462,  472,  ,473, 
498,  553,  554 

Bayer,  140 

Bfelopolsky,  166,  272 

Benzenberg,  337 

Bessel,  165 

Biela,  328,  330,  342 

Bode,  257 

Boltwood,  363 

Bond,  297,  487 

Boss,  Benjamin,  490 

Boss,  Lewis,  141,  482,  483,  488, 
491,  509,  531 

Bouguer,  42 

Bourget,  552 

Bouvard,  239 

Boys,  62 

Bradley,  95,  98 

Brandes,  337 

Braun,  62 


290, 
303, 
333, 

474, 


489, 


Bredichin,  323,  324 
Brooks,  312,  321,  327 
Brorae^,  263,  330 
BufFham,  307 
Buisson,  552 
Bunsen,  371 
Burnham,  506 

Caesar,  184 

Callisto,  151 

Campbell,   279,  482,   483,  484,  486, 

513,  515,  530,  545,  546,  553,  561 
Cannon,  Miss,  527 
Cassini,  G.  D.,  274,  275,  297,  300 
Cassini,  J.,  41,  271,  302 
Cerulli,  272 
ChaUis,  240 
Chamberlin,  73,  346,  421,  424,  425, 

437,  443,  444,  451 
Chandler,  63,  90,  91,  260,  321  . 
Chapman,  466,  467,  468,  470 
Clark,  165 
Clarke,  42 

Clerk-Maxwell,  303,  326 
Columbus,  1,  15,  40,  41 
Comstock,  483 
Condamine,  42 
Copernicus,  79,  117,  118 
Cornu,  62 
Cowell,  335 
CroU,  115 
CromeUiu,  335 
Curtis,  166 

D'Alembert,  95 

Darwin,  Charles,  16,  412,  413 

Darwin,  George  H.,  59,  63,  450,  458, 

460,  545 
Darwin,  Horace,  63 
Dawes,  506 
Delavan,  324 
Denning,  338,  340 
Deslandres,  398 
De  Vico,  330 
Doerfel,  313 


565 


566 


INDEX  OF  NAMES 


Donati,  330 

Doppler,  375,  389,  394,  397,  524 

Douglas,  291 

Dyson,  483 

Eddington,  490,  491 
Ehlert,  63 
Elkin,  514 
Ellerman,  400,  401 
Encke,  304,  329,  330 
Eratosthenes,  41 
Euclid,  116 
Eudoxus,  40,  116 
Euler,  92,  435 
Europa,  160 
Evans,  -285 
Evershed,  386 

Fabricius,  515 

Fabry,  552 

Farrington,  344,  345 

Faye,  450 

Fizeau,  292,  375,  389,  394,  397,  524 

Flamsteed,  140 

Fleming,  Mrs.,  527 

Forbes,  262 

Foucault,  84,  85 

Fowle,  350,  351 

Fox,  376,  382 

Fraunhofer,  390,  403 

Frost,  511,  513,  514,  552 

Gale,  52,  56,  59,  388,  458 

Galileo,  8,  79,  117,  119,  207,  289,  299, 

382 
Galle,  240 
Gauss,  258,  313 
Gilbert,  213 
Gill,  142,  247,  499 
Godin,  42 
Goodrieke,  515 
Gould,  139 
Gregory  XIII,  Pope,  184,  185 

Hagen,  84 

Hale,  285,  385,   386,  388,  389,  398, 

400,  401 
Hall,  273,  305 
Halley,  156,  165,  327,  332,  334,  335, 

336,  342 
Harding,  258 
Hayford,  33,  42 
Hecker,  63 
Hegel,  257 


Helmert,  42 

Helmholtz,  357,  450,  526,  533 

Hencke,  258 

Henderson,  101 

Hera,  151 

Herschel,  John,  205,  316,  467,  470, 
473,  506  ' 

Herschel,  WUliam,  239,  297,  305,  306, 
316,  329,  470,  474,  482,  505,  521, 
553,  560 
Hill,  242,  297 
Hinks,  247 

Hipparchus,  79,  94,  117,  141 
Holden,  307 
Hooke,  274,  275 
Hough,  G.  W.,  294,  296 
Hough,  S.  S.,  63 
Huggins,  279 
Hughes,  149,  157 
HuU,  326 
Hussey,  506 
Huxley,  362 
Huyghens,  297,  299 

Innes,  499 

Jacobi,  544 
Jeffreys,  91 
Joule,  355 
Julius,  396 

Kant,  357,  411,  412,  414,  416,  446, 

447,  448,  449 
Kapteyn,    142,   473,   485,    486,    490, 

491,  496,  499,  525,  526,  531 
Keeler,  279,  303,  307,  424,  554,  561 
Kelvin,  60,  62,  68,  359,  361,  405,  459, 

492 
Kepler,  7,  9,  117,  119,  229,  230,  231, 

313,  523 
Kirchhoff,  371,  390 
Kirkwood,  260,  304,  450,  451 
Kortozzi,  63 
Kilstner,  63,  90 

Lagrange,  233,  234,  238,  241 
Lambert,  313 

Lane,  357,  358,  526,  533,  540 
Langley,  350,  366,  379,  390 
Laplace,  45,  233,  238,  239,  302,  313, 

320,  411,  412,  414,  416,  449,  450, 

451,  533,  534 
Lassell,  297,  307 
Lebedew,  326 


INDEX  OF  NAMES 


567 


Lee,  514 

Leibnitz,  234 

Leverrier,  240,  241,  257,  342 

LexeU,  321 

Lindemann,  526,  527 

Lookyer,  394,  534,  555 

Love,  58 

Lowell,  262,  270,  272,  283,  284,  285, 

286,  287,  298,  304 
Ludendorff,  152 

Maclaurin,  544 

MacMiUan,  88,  459 

Magellan,  1 

Maney,  552 

Mascari,  272 

Maskelyne,  62 

Maunder,  160,  285,  384,  405 

Maury,  Miss,  527 

Mayer,  355 

Medusa,  160 

Melotte,  289,  466,  467,  468,  470 

Mendelfeeff,  369 

Messier,  156,  157,  501 

Michelson,  52,  56,  59,  292,  369,  458 

Milne,  62 

MitcheU,  62 

Moore,  553,  561 

Muller,  268,  269,  276 

Newcomb,  63,  285,  292,  308 

Newton,  7,  8,  9,  15,  34,  35,  41,  42,  62, 
63,  79,  80,  94,  119,  120,  230,  232, 
233,  238,  313,  329,  332,  355,  365, 
366,  390 

Nichols,  326 

Nicholson,  289 

Olbers,  258,  323 
Olivier,  338,  340 
Orloff,  63 

Parkhurst,  259 

Perrine,  289,  525 

Perrotin,  272,  284,  307 

Philolaus,  78 

Piazzi,  257,  258 

Picard,  41,  42 

Pickering,  E.  C,  152,  470,  512,  524, 

527 
Pickering,  W.  H.,  216,  262,  284,  287, 

297,  319 
Poincarg,  242,  544 
Poisson,  459 


Ptolemy,  79,  117,  U8,  139,  141 
Pythagoras,  40,  116 

Ramsay,  395 

Rayet,  530,  535,  561 

Rebeur-Pasohwitz,  63 

Reich,  62 

Ritchey,  210,  402,  429,  430,  501,  525, 

537,  551,  556,  559 
Ritter,  357 

Roche,  303,  327,  346,  423,  450 
Roemer,  292 
Rowland,  369,  385,  390 
Russell,  617,  534 
Rutherford,  367 

Sampson,  390 

Schaeberle,  166 

Schiaparelli,  270,  272,  283,  284,  285, 

342 
Schlesinger,  518 
Schroter,  269,  271 
Schuster,  385 
Schwabe,  383 
Schwarzschild,  326 
Schweydar,  58,  63 
Seares,  555 
Secchi,  527,  528 
See,  507,  508 
Seeliger,  525 
Shapley,  503,  517,  519 
Slipher,  E.  C,  295,  298 
Slipher,  V.  M.,  272,  279,  307,  308, 

558,  561 
Slocum,  397,  426 
Smith,  389 
Sosigenes,  184 
Spencer,  16,  412 
Stefan_,.J280,  354,  358,  542 
SOohn,  386,  387,  394 
Stromgren,  314 
Strutt,  363 
Struve,  William,  506 
SuUivan,  560 
Sundman,  242 

Tacchini,  272 
Tebbutt,  331 
Tempel,  342 
Thackeray,  483 
Thales,  116 
Thetis,  151 
ThoUou,  284 
Tisserand,  308 


568 


INDEX  OF  NAMES 


Titius,  257 
Todd,  262 
Turner,  467 
Tuttle,  342 

Tyoho  Brahe,  7,  118,  119,  141,  153, 
229,  523 

Very,  206 

Vogel,  279,  512,  513,  518 

Wallace,  Alfred  Russel,  412 
WaUace,  R.  J.,  161,  215 
Weiss,  342 
WheweU,  234 
Wien,  372 
Wilczynski,  390 


Williams,  284,  524 

Wilsing,  62,  390 

Wilson,  R.  E.,  553 

Wilson,  W.  E.,  525 

Witt,  247,  260 

Wolf,  Max,  258,  525 

Wolf,  530,  535,  561 

Wollaston,  390,  487 

Wright,  Thomas,  411,  412,  446 

Wright,  W.  H.,  515 

Young,  C.  A.,  307,  391 
Young,  Thomas,  365 

Zeeman,  385 
Zeus,  151,  160 
ZoUner,  205,  487 


GENERAL  INDEX 


Absorption  of  light,  350,  467. 

Absorption  spectrum,  375. 

Acceleration,  definition  of,  8. 

Achernar,  144. 

Aerolites,  343. 

Age  of  earth,  360. 

Alcor,  151,  514. 

Aldebaran,  139,  144,  521. 

Algol,  140, 159, 160, 166, 515, 517, 518. 

Almagest,  117. 

Almucantars,  124. 

Alpha  Centauri,  101,  144,  476,  515. 

Alpha  Crucis,  144. 

Alpha  Geminorum,  519. 

Altair,  139,  144. 

Altitude,  124. 

of  equator,  108. 

of  pole,  108. 
American   Ephemeris   and"    Nautical 

Almanac,  176,  253. 
Andromeda,  159,  160. 

Nebula,  158,  556,  558,  559. 
Andromid  meteors,  340, 341,  342,  346. 
Angular  distances,  150. 
Antares,  144,  156,  503,  529. 
Aphelion  point,  104. 
Apogee,  197. 
Aquarid  meteors,  342. 
Aquila,  473. 
Ara,  473. 

Arcturus,  144, 157,  486,  503,  528,  529. 
Areas,  law  of,  104,  229. 
Argo,  473. 

Ascending  node,  188,  249. 
Astronomical  unit,  227. 
Atmosphere,  64. 

absorption  of  light  by,  350. 

climatic  influences  of,  71. 

composition  of,  64. 

height  of,  66. 

mass  of,  65. 

of  Jupiter,  296. 

of  Mars,  276. 

of  Mercury  and  Venus,  268. 

of  Moon,  203. 


of  Saturn,  306. 

of  Uranus  and  Neptune,  307. 

pressure  of,  65. 

refraction  by,  74. 

r61e  of  in  life  processes,  74. 
Atoms,  68. 
August  meteors,  342. 
Auriga,  160. 
Aurorse,  66,  404. 
Autumnal  equinox,  109. 
Azimuth,  124. 

Base  line,  30. 

Beehive  (Prsesepe),  166. 

Belt  of  Orion,  163,  165. 

Beta  Amigae,  490. 

Beta  Centauri,'  144. 

Beta  Geminorum,  528. 

Beta  Lyri,  155,  515,  518. 

Betelgeuze,  144,  162,  165,  523. 

Biela's  comet,  328,  330,  342. 

Big  Dipper,  77,  139,  140,   149,  151, 

153,  160,  488,  490,  527. 
Binary  stars,  507. 

evolution  of,  543. 

masses  of,  508. 

orbits  of,  507. 

origin  of,  543. 

spectroscopic,  510. 
Bode's  law,  257. 
Bolometer,  366. 
Bootes,  157, 

Brooks'  comet,  318,  321. 
Brorsen's  comet,  330. 

Calendar,  184. 

Canals  of  Mars,  283. 

Canes  Venatiei,  spiral  nebula  in,  429. 

Canis  Major,  165,  473. 

Canis  Minor,  165. 

Canopus,  144,  480. 

CapeUa,  144,  160,  486,  514,  524. 

Carbon  dioxide,  64. 

effects  on  climate,  73. 

production  of,  73. 


569 


570 


GENERAL  INDEX 


Cassiopeia,  152, 153, 159,473,474,523. 

Castor,  166. 

Catalogues  of  stars,  141,  482,  499. 

Celestial  sphere,  122. 

Centaurus,  473. 

Center  of  gravity  of  eartli  and  moon, 

199. 
Cepheus,  473. 
Ceres,  discovery  of,  257. 
Chemical  constitution  of  sun,  393. 
Chromosphere,  378,  394. 
Circinus,  473. 
Circumpolar  star  trails,  78. 
Clusters  of  stars,  500. 
Comet,  of  1668,  318. 

of  1680,  329. 

of  1811,  316,  329. 

of  1843,  318. 

of  1880  and  1882,  318,  331. 
Comets,  appearance  of,  311. 

capture  of,  320. 

dimensions  of,'316. 

disintegration  of,  327. 

families  of,  318.  ' 

masses  of,  317. 

naming  of,  313. 

orbits  of,  313. 

origin  of,  322,  442. 
Comets'  tails,  theories  of,  323. 
Conic  sections,  234,  313. 
Conservation  of  energy,  355. 
Constellations,  139. 

list  of,  148. 
Contraction  of  sun,  356. 
Coordinates,  123. 
Copernican  theory,  118. 
Corona,  of  sun,  379,  401. 
Corona  Borealis,  157. 
Coronium,  403. 
Corpuscles,  367. 
Craters  of  moon,  211. 
Crux,  473. 
Cygnus,  473,  550,  551,  554. 

Date,  place  of  change  of,  181. 
Day,  astronomical,  181. 

civil,  181. 

invariability  of,  88. 

Julian,  185. 

longest  and  shortest,  173. 

mean  solar,  175. 

sidereal,  171. 

solar,  172. 
Dearborn  Observatory,  165. 


Declination,  126. 

Deduction,  9,  10. 

Deferent,  118. 

Deimos,  273. 

Delavan's  comet,  324,  325. 

Delta  Aquila),  508. 

Delta  Cephei,  520,  522. 

Delta  Librse,  518. 

Deneb,  144. 

Density,  of  earth,  45,  46,  48,  50. 

of  moon,  202,  254. 

of  sun,  254. 

of  stars,  517,  541. 
Deviation,  of  falling  bodies,  82. 

of  air  currents,  85. 

of  rivers,  87. 
Dialogues  of  Galileo,  119. 
Dimensions,  of  comets,  316. 

of  sun,  moon,  and  planets,  254. 
Discovery  of    Uranus  and  JJeptune, 

155,  238. 
Disintegration,  of  comets,  327. 

of  matter,  363,  364. 
Distance,  of  moon,  20,  194. 

of  planets,  249. 

of  stars,  476,  484,  486,  487. 

of  sun,  247. 
Distribution,  of  stars,  463,  470. 

of  sun  spots,  383. 

of  time,  179. 
Diurnal  circles,  109. 
Donati's  comet,  330. 
Doppler-Fizeau  law,   375,   389,   394, 

397,  524. 
Double  stars,  505. 
Dynamics  of  stellar  system,  491. 

Earth,  age  of,  360. 

density  of,  45,  46,  48,  50. 

dimensions  of,  33. 

elasticity  of,  59. 

mass  of,  45. 

oblateness  of,  31,  34,  35. 

pressure  in,  51. 

revolution  of,  96. 

rigidity  of,  52,  59. 

rotation  of,  77,  82,  84,  85. 

sphericity  of,  27. 

temperature  in,  51. 
Earthquakes,  60. 
Earth's  orbit,  103,  104. 
Eccentricity,  1(04. 

of  earth's  orbit,  104,  249. 

of  planetary  orbits,  249. 


GENERAL  INDEX 


571 


Eccentric  motion,  118. 
Echelon  spectroscope,  369. 
Eclipses,  of  moon,  218. 

of  sun,  220. 

phenomena  of,  223. 

uses  of,  220,,22i- 
Eclipsing  variables,  516. 
Ecliptic,  94,  127. 

obliquity  of,  105. 

pole  of,  106. 
Elasticity  of  earth,  52,  59. 
Electrical  repulsion,  323. 
Electrons,  367. 
Elements,  in  sun,  393. 

of  orbit,  248,  249. 

table  of,  249. 
Eleven-year  cycle,  404. 
Ellipse,  definition  of,  103. 
Elongations  of  planets,  227. 

dates  of,  256. 
Encke's  comet,  329,  330. 
Energy,  conservation  of,  355. 

from  radium,  363. 

kinetic,  356. 

of  coal,  353. 

of  solar  system,  419. 

of  water,  352. 

of  wind,  352. 

potential,  356. 

radiant,  356. 

radiated  by  sun,  353. 
Epicycle,  118. 

Epsilon  Lyra,  154,  155,  239. 
Equation  of  time,  176. 
Equator,  106,  125. 

altitude  of,  108. 
Equinoctial  colure,  125. 
Equinoxes,  94,  109. 

autumnal,  109. 

how  to  locate,  153. 

precession  of,  92,  94,  115. 

vernal,  109. 
Eros,  260. 

Escape  of  atmosphere,  69. 
Eta  Cassiopeiae,  153. 
Evolution,  16. 

essence  of,  407. 

of  planets,  431. 

of  stars,  532,  533. 

value  of,  408. 

Faoulae,  382. 

periodicity  of,  388,  404. 
Falling  bodies,  deviations  of,  82. 


First-magnitude  stars,  143,  144. 
Flash  spectrum,  391. 
Flocculi,  389. 
Foci,  103. 
Fomalhaut,  144. 
Fossils,  occurrence  of,  362. 
Fouoault's  pendulum,  84. 
Fraunhofer  lines,  390. 

Galaxy,  146,  159,  470,  474,  479,  490, 
492,  496,  497,  498,  499,  500,  503, 
553,  558. 
Galileo's  Dialogues,  119. 
Gamma  Virginis,  508. 
Gases,  kinetic  theory  of,  68,  492. 

pressure  of,  69. 
Gegenschein,  262. 
Gemini,  166. 

Geographical  system,  122. 
Glacial  epoch,  73. 
Globular  star  clusters,  500. 
Grating  spectroscope,  369. 
Gravitation,  discovery  of,  230. 

importance  of  law  of,  231. 

law  of,  9,  230,  463. 
Gravity,  surface,  245. 

of  planets,  254. 

Halley's  comet,  327,  332,  334,  335, 

336,  342. 
Harvard   College  Observatory,   144, 
260,  512,  522,  523,  527,  528,  529, 
530. 
Heat,  from  moon,  204. 
from  sun,  350. 
received  by  planets,  250. 
Hehum,  362,  363,  395. 
Hercules,  156,  159,  482,  501,  503. 
Horizon,  123. 
Hour  angle,  131. 
Hour  circle,  125. 
Hyades,  160,  162,  488. 
Hydrocyanic  acid,  64. 
Hyperbola,  235. 
Hypothesis,  of  Kant,  446. 
of  Laplace,  449. 
planetesimal,  421. 

Inclination  of  earth's  orbit,  105. 

of  planetary  orbits,  249. 
Induction,  8. 

Infinity  of  physical  universe,  548. 
Irregular  nebulse,  550. 

variables,  522. 
Isostasy,  42. 


572 


GENERAL  INDEX 


Juno,  discovery  of,  258. 
Jupiter,  atmosphere  of,  296. 

belts  of,  293. 

great  red  spot  on,  294. 

markings  on,  293. 

physical  condition  of,  296. 

rotation  of,  292,  437. 

satellite  system  of,  289. 

seasons  of,  296. 

Kepler's  laws,  229. 

Kinetic  energy,  356. 

Kinetic  theory  of  gases,  68,  492. 

Lag  of  tides,  455. 
Lane's  law,  358,  526. 

paradox,  357,  533. 
Laplacian  hypothesis,  449,  533. 
Latitude,  astronomical,  40,  123. 

celestial,  127. 

geocentric,  40. 

geographical,  40. 

variation  of,  63,  89. 
Law,  of  areas,  104,  229. 

of  gravitation,  9,  230,  463. 
Laws,  of  force,  236. 

of  motion,  8,  80. 

of  spectrum  analysis,  371. 
Leap  year,  184. 
Leo,  157,  340. 

Leonid  meteors,  340,  341,  342. 
Lexell's  comet,  321. 
Libration  of  Mercury,  271. 
Librations  of  moon,  201. 
Lick  Observatory,  150,  160,  166,  260, 
277,  278,  285,  289,  291,  424,  483, 
507,  515,  530,  553,  555,  557,  561. 
Light,  absorption  of,  370. 

dispersion  of,  370. 

from  moon,  204. 

from  sun,  349. 

nature  of,  365. 

polarized,  366. 

pressure  of,  326. 

production  of,  366. 

refraction  of,  74,  370. 

velocity  of,  22,  99,  291,  354. 

wave  lengths  of,  349,  366. 

zodiacal,  262,  328,  442. 
Longitude,  123. 

celestial,  127. 
Long  period  variables,  520. 
Lowell  Observatory,  272,    279,   285, 
295,  308. 


Lunar,  craters,  211. 

mountains,  207. 
Lupus,  473. 
Lyra,  23,  153,  156. 
Lyrid  meteors,  341,  346. 

Magellanic  clouds,  530,  553. 
Magnetic  storms,  periodicity  of,  404, 

405. 
Magnitudes  of  stars,  142,  465. 
Mars,  atmosphere  of,  276. 

canals  of,  283. 

explanation  of  canals  of,  285. 

polar  caps  of,  277,  278. 

rotation  of,  274,  437. 

satellites  of,  273. 

seasons  of,  277. 

temperature  of,  277. 

water  on,  279. 
Mass,  of  atmosphere,  65. 

of  moon,  71,  198. 

of  sun,  254. 
Masses,  determination  of,  244. 

of  planets,  254. 

of  stars,  508,  509. 
Mean  distance,  definition  of,  229. 
Mean  solar  time,  175. 
Mercury,  266. 

albedo  of,  268. 

atmosphere  of,  268. 

librations  of,  271. 

markings  of,  269. 

phases  of,  266. 

rotation  of,  269. 

seasons  of,  270. 

transits  of,  267. 
Meridian,  124. 
Meteoric  showers,  339. 

matter,  resistance  of,  88. 
Meteorites,  343. 

composition  of,  344. 

origin  of,  345. 
Meteors,  65,  337,  525. 

effects  of  on  earth's  rotation,  88. 

effects  of  on  solar  system,  343. 

height  of,  65,  338. 

number  of,  338. 
Mile,  nautical,  16. 

MUky  Way,  22,  146,  160,   431,  462, 
470,  473,  490,  491,  496,  498,  507, 
523,  525,  530,  531,  554,  557,  558, 
560. 
Mizar,  151,  152,  612. 

spectrum  of,  511,  513. 


GENERAL  INDEX 


573 


Molecules,  68. 

size  of,  68. 

velocity  of,  69. 
Moment  of  momentum,  88. 

of  solar  system,  416,  417. 
Monooeros,  473. 
Moon,  188. 

apogee  of,  197. 

apparent  motion  of,  188. 

atmosphere  of.  203  j 

craters  of,  211. 

density  of,  202,  254. 

dimensions  of,  196. 

distance  of,  20,  194. 

diurnal  circles  of,  192. 

eclipses  of,  218. 

effects  of  on  earth,  217. 

heat  received  from,  204. 

librations  of,  201. 

map  of,  209. 

mass  of,  71,  198,  254. 

mountains  of,  207. 

orbit  of,  188,  197. 

perigee  of,  197. 

periods  of,  189. 

phases  of,  191. 

rays  and  rills  of,  214. 

rotation  of,  200. 

satellites  of,  220. 

surface  changes  of,  216. 

surface  gravity  of,  202. 

temperature  of,  205. 

velocity  of,  196. 
Motion,  of  earth,  103. 

of  sun,  96,  482,  483,  484. 

of  stars,  145,  480,  481,  487. 
Mount    Wilson    Solar    Observatory, 
285,  348,  387,  396,  401,  501,  503, 
554. 
Mountain     method    of    determining 

density  of  earth,  48. 
Mu  Orionis,  512. 

spectrum  of,  513. 
Musoa,  473.  ' 

Nadir,  124. 

Naval  Observatory,  17,  123,  180,  181. 

Nebulae,  irregular,  550. 

planetary,  560.  ' 

ring,  560. 

spiral,  429,  430,  554,  556,  557. 
Nebular  hypothesis,  411,  449. 
Neptune,  atmosphere  of,  307. 

discovery  of,  155,  238. 


physical  condition  of,  308. 

rotation  of,  307,  437. 

satellite  of,  306. 
Nitrogen,  64. 
Nodes,    ascending    and    descending, 

188. 
Norma,  473. 
Northern  Crown,  157. 
Nova  Aurigae,  524. 
Nova  Persei,  525,  526. 
Number  of  stars,  145,  464,  466,  468. 
Nutation,  95. 

Oblate  figure,  32. 
Oblateness  of  earth,  31,  34. 
Obliquity  of  ecliptic,  105. 
Omega  Ceutauri,  501,  522. 
Omicron  Ceti,  515,  521. 
Ophiuchus,  473,  523. 
Opposition,  definition  of,  228. 

of  planets,  dates  of,  256. 
Orbits,  of  binary  stars,  507. 

of  comets,  313. 

of  planetoids,  259. 

of  planets,  elements  of,  248,  249. 
Origin,  of  binary  stars,  543. 

of  comets,  322,  442. 

of  meteorites,  345. 

of  planetoids,  259. 

of  planets,  431. 

of  species,  412,  413. 

of  spiral  nebulae,  424. 
Orion,  77,  160,  162,  163,  491. 
Orion  nebula,  163,  164,  552. 
Orionid  meteors,  341. 
Oxygen,  64. 

Pallas,  discovery  of,  258. 

Parabola,  235. 

Parallax,  of  stars,  definition  of,  100. 

determination  of,  476. 

of  sun,  247. 
Parallelogram  of  forces,  81. 
Parsec,  definition  of,  476. 
Pendulum,  Foucault's,  84. 

horizontal,  60,  63. 
Penumbra,  of  earth's  shadow,  218. 

of  sun  spots,  381. 
Perigee  of  moon's  orbit,  197. 
Perihelion  point,  definition  of,  104. 

longitude  of,  249. 
Period,  of  moon,  sidereal,  189. 

synodical,  189. 
Period  of  planets,  249. 


574 


GENERAL   INDEX 


Periodicity  of  sun  spots,  383. 

Perseid  meteors,  340. 

Perseus,  140,  159,  160,  473,  490,  523. 

Perturbations,  237. 

Phases,  of  Mercury  and  Venus,  266. 

of  moon,  191. 
Phobos,  273. 

Photographic  chart  of  sky,  141. 
Photosphere,  378,  379. 
Planetary  orbits,  dimensions  of,  249. 

eccentricities  of,  249,  434. 

planes  of,  249,  433. 
Planetesimal,  hypothesis,  421. 

organization,  422. 
Planetoids,  diameters  of,  260. 

orbits  of,  259,  442. 

origin  of,  259. 
Planets,  226. 

dates  of  elongation  of,  256. 

dates  of  opposition  of,  256. 

density  of,  254. 

dimensions  of,  254. 

distances  of,  249. 

evolution  of,  431. 

heat  received  by,  250. 

inferior,  227. 

intra-Mercurian,  261. 

masses  of,  254. 

origin  of,  431. 

periods  of,  249. 

possible  undiscovered,  261. 

rotations  of,  437. 

superior,  227. 

surface  gravity  of,  254. 

synodical  periods  of,  256. 

trans-Neptunian,  261. 
Pleiades,  22,  139,  160,  161,  162,  536, 

537,  541. 
Pointers,  149,  150. 
Pole,  106. 

altitude  of,  108. 

of  ecliptic,  106. 
Polar  caps  of  Mars,  277,  278. 
Polaris,  139,  149,  150,  153,  515. 
Pollux,  144,  166. 
Potential  energy,  356. 
Praesepe,  166. 

Precession  of  equinoxes,  92,  94,  115. 
Principia,  232. 
Prism  spectroscope,  369. 
Procyon,  144,  165,  166. 
Prominences,  379,  395,  426. 
Proper    motion    of    stars,    146,   479, 

498. 


Ptolemaic  theory,  118. 
Pulkowa,  166. 
Pyramids,  23. 

Quadrature,  191,  228. 

Radial  velocity,  144,  375,  377. 

Radiant  point  of  meteors,  339,  341. 

Radioactivity  in  sun,  363. 

Radium,  362,  363. 

Rays  and  rills,  214. 

Reference  points  and  lines,  121. 

Refraction,  74,  370. 

Regulus,  144,  159. 

Reversing  layer,  378,  390. 

constitution  of,  392. 
Revolution   of    earth,    96,    98,    100, 

101. 
Rigel,  144,  163,  480. 
Right  ascension,  126. 
Rigidity  of  earth,  52,  59. 
Ring  nebula  in  LjTa,  155,  560. 
Rings  of  Saturn,  299,  441. 

constitution  of,  302. 

permanency  of,  304. 
Roche's  limit,  303,  327,  346,  450. 
Rotation,  of  earth,  82,  84,  85. 

of  Jupiter,  292,  437. 

of  Mars,  274,  437. 

of  Mercury,  269. 

of  moon,  200. 

of  Neptune,  307,  437. 

of  Saturn,  305,  437. 

of  sun,  388,  436. 

of  Uranus,  307,  437. 

of  Venus,  271. 
Runaway  stars,  498. 

Sagittarius,  473,  554, 
Salinity  of  the  oceans,  361. 
Satellites,  of  Jupiter,  289. 

of  Mars,  273. 

of  moon,  220. 

of  Neptune,  306. 

of  Saturn,  297. 

of  Uranus,  306. 

origin  of,  440. 
Saturn,  physical  condition  of,  306. 

ring  system  of,  299,  441. 

rotation  of,  305,  437. 

satellite  system  of,  297. 

seasons  of,  306. 

shape  of,  39. 

surface  markings  on,  305. 


GENERAL  INDEX 


575 


Science,  1. 

imperfections  of,  10. 

methods  of,  6. 

origin  of,  4. 

value  of,  2. 
Scientific  theories,  12. 

contributions  to,  by  astronomy,  14. 
Scintillation  of  stars,  76. 
Scope  of  astronomy,  19. 
Scorpius,  156,  157,  473. 
Seasons,  cause  of,  l07. 

lag  of,  112. 

length  of,  112. 

of  Jupiter,  296. 

of  Mars,  277. 

of  Mercury,  270. 

of  Saturn,  306. 

of  Venus,  272.  ' 

Seismograph,  62. 
Serpens,  473. 
Shape  of  earth,  33,  38. 
Shape  of  earth's  orbit,  102. 
Shooting  stars,  65,  337,  525. 
Sidereal,  day,  171. 

period  of  moon,  189. f- 

period  of  planets,  249. 

year,  183. 

time,  171. 
Siderites,  342. 

Sirius,  139,  140,    143,  144,  165,  166, 
322,  479,  480,  486,  488,  493,  494. 

spectrum  of,  527. 
Solar,  days,  172. 

energy,  353. 

Observatory,   285,   348,    386,   387, 
396,  401,  501,  503,  554. 

time,  172. 
Solstices,  109. 

Spectra  of  stars,  486,  527,  530. 
Spectroheliograph,  385,  398. 
Spectroscope,  101,  269,  279,  303,  307, 

369,  463. 
Spectroscopic  binaries,  510. 
Spectrum,  absorption,  375. 

analysis,  369. 

analysis,  laws  of,  371. 

flash,  391.      ' 
Sphericity  of  earth,  27. 
Spheroid,  oblate  and  prolate,  38. 
Spica,  144,  153,  514. 
Spiral  nebvdffi,  429,  430,  554,  556,  557. 

origin  of,  424. 
Stability,  of  solar  system,  238. 

of  satelUtes,  299. 


Standard  time,  177. 
Star,  clusters,  500. 

streams,  490. 
Stars,  binary,  507. 

catalogues  of,  141,  482,  499. 

clusters  of,  500. 

density  of,  517,  541. 

distances  of,  476,  484,  486,  487. 

distribution  of,  463,  470. 

double,  605. 

evolution  of,  532,  533. 

first-magnitude,  143,  144. 

groups  of,  487,  499. 

masses  of,  508,  509. 

motions  of,  145,  480,  481. 

number  of,  145,  464,  466,  468. 

parallaxes  of,  476. 

proper  motions  of,  146,  479,  481. 

radial  velocities  of,  481. 

runaway,  498. 

spectra  of,  486,  527. 

temperatvires  of,  539. 

temporary,  523. 

twinkling  of,  76. 

variable,  515. 

velocities  of,  23. 
Stefan's  law,  280,  354,  358. 
Sun,  apparent  motion  of,  96. 

constitution  of,  378,  392,  393. 

density  of,  254. 

distance  of,  247. 

eclipses  of,  220. 

heat  received  from,  350. 

light  and  heat  of,  349. 

magnetic  field  of,  385. 

magnitude  of,  143,  602. 

mass  of,  254. 

motion  of,  482,  483,  484. 

parallax  of,  247. 

past  and  future  of,  360,  443. 

radiation  of,  353. 

rotation  of,  388,  436. 

surface  gravity  of,  245,  254. 

temperature  of,  354. 
Sunlight  in  all  latitudes.  111. 
Sun's  eleven-year  cycle,  404. 
Sun's   heat,    combustion    theory   of, 
358. 

contraction  theory  of,  356. 

meteoric  theory  of,  358. 

sub-atomic  energy  theory  of,  364. 
Sun   spots,    distribution   and   perio- 
dicity of,  383. 

motions  of,  387. 


576 


GENERAL  INDEX 


Sun  spots,  penumbra  of,  381. 

periodicity  of,  383. 

polarity  of,  386. 

umbrae  of,  381. 
Superstition,  14. 

Surface    gravity,    determination     of, 
245. 

of  moon,  202. 

of  planets,  254. 

of  sun,  245,  254. 
Sword  of  Orion,  163. 
Synodioal  period,  of  moon,  189. 

of  planets,  256. 

Tails  of  comets,  theories  of,  323. 
Taurus,    160,    162,    473,    488,    489, 

494. 
Tebbutt's  comet,  331. 
Tempel's  comet  of  1866,  342. 
Temperature,  of  earth,  51. 

of  Mars,  277. 

of  moon,  205. 

of  sun,  354. 

of  stars,  539. 
Temporary  stars,  523. 
Theory  of  evolution,  407. 

value  of,  408. 
Tidal,  bulges,  54. 

cones,  455. 

evolution,  420,  454,  460. 

experiments,  56. 
Tide-raising,  acceleration,  54. 

forces,  243,  452,  453. 
Tides,  cause  of,  242. 

eifects  of,  on  day,  88. 

effects  of,  on  earth,  458. 

effects  of,  on  moon,  456. 

lag  of,  455. 
Time,  distribution  of,  179. 

equal  intervals  of,  169,  170. 

equation  of,  176. 

local,  177. 

mean  solar,  175. 

practical  measure  of,  170. 

sidereal,  171. 

solar,  172. 

standard,  177. 
Torsion  balance,  46. 
Total  eclipses,  222. 
Transits    of     Mercury    and    Venus, 

267. 
Triangulation,  29. 
Trifid  Nebula,  554,  555. 
Tropical  year,  183. 


Tuttle's  comet,  342. 
Twilight,  duration  of,  67. 
Twinkling  of  stars,  76. 

Umbra,  of  earth's  shadow,  218. 

of  sun  spots,  381. 
Uniformity  of  earth's  rotation,  87. 
Uranium,  362. 
Uranus,  atmosphere  of,  307. 

discovery  of,  239. 

physical  condition  of,  308. 

rotation  of,  307,  437. 

satellites  of,  306. 
Ursa  Major,  150,  151,  490,  556. 

Variability,  of  Eros,  261. 

of  Japetus,  297. 
Variable  stars,  cluster,  522. 

eclipsing,  516. 

irregular,  522. 

of  Beta  Lyrse  type,  518. 

of  Delta  Cephei  type,  519. 

long  period,  520. 
Variation,  in  lengths  of  days,  172. 

of  latitude,  63,  89. 

of  sun's  radiation,  351. 
Vega,  23,  139,  144,  154,  156,  486. 
Velocity,  of  escape,  69. 

of  light,  22,  99,  291,  354. 

of  meteors,  337. 

of  molecules,  69. 

of  moon,  196. 

of  sun,  23. 

of  stars,  23. 
Venus,  atmosphere  of,  268. 

markings  of,  271. 

phases  of,  266. 

rotation  of,  271. 

seasons  of,  272. 

transits  of,  267. 
Vernal  equinox,  109. 
Vertical  circles,  124. 
Vesta,  discovery  of,  258. 
Virgo,  153. 
Vulpecula,  473. 

Wave  length  of  light,  349,  366. 

Wien's  law,  372. 

Wolf-Rayet  stars,  530,  535,  561. 

Xeon,  64. 

Year,  anomalistic,  183. 
leap,  184. 


GENERAL  INDEX 


577 


Year,  sidereal,  183. 
tropical,  183. 

Yerkes  Observatory,  77,  139,  158, 
161,  163,  164,  192,  208,  210,  212, 
215,  259,  275,  285,  291,  300,  301, 
302,  312,  333,  376,  397,  400,  426, 


429,  4.30,  462,  501,  511,  513,  525, 
528,  529,  537,  551,  554,  556,  559. 

Zenith,  124. 

Zodiacal  light,  262,  328,  442. 


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